After the Oilers were eliminated in the 1982 NHL playoffs, Kurri was able to join the Finnish national team in 1982 World Championships. The tournament was held in Finland. Kurri formed Finland’s top line along with SM-liiga players Reijo Leppänen of TPS and Kari Jalonen of Kärpät. The lineup led the Finnish team in points and played well during the tournament but Finland was not able to advance to the medal round.

Kurri’s 19 goals in the 1985 post-season (18 games played) tied an NHL record set by the Flyers’ Reggie Leach in the 1976 playoffs (16 games played). Those 19 goals included four hat tricks, a still-standing record in a playoff season. Kurri scored three of the hat tricks against the Blackhawks in the conference finals, also a feat that no player has ever equaled. He scored five goals in a single game, on November 19, 1983. The achievement was not repeated by Finnish player until November 24, 2018 when Patrik Laine had a five-goal-game. Kurri registered three goals and two assists in Game 2 of the Stanley Cup Finals in Boston, May 18, 1990, setting a record for scoring in a single Finals game in the modern era, on his 30th birthday. Kurri had also set up Petr Klíma’s goal in triple overtime in Game 1.

Kurri won five Stanley Cups, all with Edmonton (1984, 1985, 1987, 1988 and 1990). In 1984–85 he had 135 points and scored 50 goals in his first 50 games; however, his 50th game was the Oilers’ 53rd game so it does not count as an official 50 goals in 50 games. A year later, he led the league in goals with 68. In 1984–85 Kurri set a record for goals by a right winger when he scored 71, which was broken by Brett Hull when he scored 72 goals in 1989–90 (Hull’s 86 goals in 1990–91 is the current record). Kurri also scored the Stanley Cup winning goal in game seven against the Philadelphia Flyers in 1987.

During his career in Edmonton, he was nicknamed the “Finnish Flash” (a title since bestowed on Teemu Selänne). Kurri was “by far our most complete player”, according to Oilers’ director of personnel Barry Fraser. Although Kurri never won the Selke Trophy, he was regarded as one of the best defensive forwards in the NHL. Alongside Kurri, Esa Tikkanen and Gretzky, Edmonton had future hall-of-famers Paul Coffey, Grant Fuhr, Glenn Anderson and Mark Messier. The team won four Stanley Cups during Gretzky’s tenure. In 1988, Gretzky was traded, along with team enforcer Marty McSorley, to the Los Angeles Kings. Following Gretzky’s trade, Kurri recorded 195 points in 154 games, leading the Oilers to their fifth and last Stanley Cup in 1990.

Kurri’s contract with the Oilers expired in 1990 but his NHL rights were still owned by the Oilers. Instead of accepting a new contract from the Oilers, Kurri opted for a season in Italy and played for Devils Milano of the Italian Serie A. Kurri had 27 goals and 48 assists in 30 games while he played in the Italian league.

In 1991, Kurri represented Finland at the 1991 World Championships held again in Finland. Finland ranked 5th of 8 teams. Later that year, at the 1991 Canada Cup, Finland lost 3–7 to the United States in the semi-finals.

After a season in Italy, Kurri’s rights were traded to the Philadelphia Flyers and then to the Los Angeles Kings on the same day. For the first time since the 1988 trade, Kurri was reunited with Wayne Gretzky. The pair had a fairly successful campaign in 1993 leading the Los Angeles Kings to the Stanley Cup Finals where they faced off against the Montreal Canadiens. Kurri managed to put together 17 points in 24 games during the run, in which the Kings ultimately fell short and lost the series in 5 games. Jari followed the run up with a few more solid seasons as a Los Angeles King, scoring 146 points in 176 games; his decline in scoring was primarily due to his transition into a more defensive minded forward in a somewhat reduced role. (Though they were back on the same team, by this time he and Gretzky were not usually playing on the same line.)

During the 1994–95 NHL lockout Jari Kurri represented Jokerit, the team for which he played before his NHL career. Kurri played alongside Teemu Selänne, the 1993 winner of the Calder Memorial Trophy who had established himself as one of the top stars in Finland during early success in the NHL. Kurri had one memorable moment during his visit to SM-liiga when Jokerit faced their local rivals HIFK. Kurri played against his former Oilers-linemate Esa Tikkanen and another Finnish NHL-player, Christian Ruuttu. Kurri, Selänne and Jokerit captain Waltteri Immonen led Jokerit to 4–2 victory over Russian HC Lada Togliatti in 1994 European Cup Finals.

At the 1994 World Championships, Kurri managed to get his first international medal when Finland earned silver after losing 1–2 in a shootout against Canada.

After the lockout Kurri returned to Los Angeles, but was traded to New York Rangers in 1996 for their playoff run.

At the 1998 Winter Olympics, Kurri faced Gretzky on ice for the last time, when Finland and Canada played for the Olympic bronze medal in Nagano. Finland was the underdog against Canada, but Finland won the game 3–2. Kurri scored the first goal of the game, which turned out to be his last goal for the Finnish national team.

Kurri is married to former Miss Finland, Vanessa Kurri, nee Forsman; they wed in 2004 and have three daughters: Odessa (born 2002), Alissa (born 2005) and Isla (2012), and one son, Paulus (born 2007). Kurri also has twin boys (Joonas and Ville) from his first marriage, to Tiina Kurri. Former linemate Wayne Gretzky is the godfather of Kurri’s children from his first marriage.

Kurri finished his career as the highest scoring European-born-and-trained player in NHL history, with 601 goals, 797 assists, and 1,398 points. (Stan Mikita, who was born in Europe, but raised in Canada, retired 18 seasons before Kurri with 926 assists and 1,467 points.) Kurri has since been passed in all three categories by Jaromír Jágr, who also passed all of Mikita’s marks. Kurri has since also been bested in goals by Teemu Selänne (the current record-holder for NHL goals scored by a Finn), and in assists by Nicklas Lidström. He was also the highest ranked non-Canadian in NHL points and goals; the only non-Canadian with more assists at Kurri’s retirement was American Phil Housley, since passed by Mike Modano. Kurri finished with 106 career playoff goals and 233 playoff points, third all-time behind Gretzky and Messier. His 1,398 points are 19th all-time, and the third highest among players trained in Europe behind Jágr and Selänne. He is 18th in all-time goals with 601, and was the leading Finn in that department until Selänne passed him in 2010. He is the second leading point-scorer in Oilers history, with 1,043 points in 754 games, behind Gretzky. His No. 17 has been retired by the Edmonton Oilers, the Finnish national team, and Jokerit. In addition, the Jari Kurri trophy is given to the most valuable player in the SM-liiga playoffs. In 2000, he was named along with longtime linemate Gretzky to the International Ice Hockey Federation Hall of Fame. In 1998, he was ranked number 50 on The Hockey News’ list of the 100 Greatest Hockey Players, the highest-ranking Northern European player (1998). In addition, he was the first Finn to be inducted into the Hockey Hall of Fame, doing so in 2001.