On 9 April 2019, Son scored the very first European competition goal at the Tottenham Hotspur Stadium in a 1–0 win against Manchester City in the quarterfinals of the 2018–19 Champions League. In the return leg, Son scored twice to earn Tottenham a 4–4 aggregate win on the away goals rule and help the club reach the semi-finals of the competition for the first time since 1962, and only the second time ever. His brace also saw him become the highest scoring Asian player in the history of the tournament with 12 goals, overtaking the previous record holder, Maxim Shatskikh. On 4 May 2019, Son received his first red card in the Premier League due to his vindictive action against Jefferson Lerma.
Son opened his 2019–20 account on 14 September 2019 by scoring two goals against Crystal Palace in the Premier League with a final result of 4–0. On 21 October, Son was named in the 30-man shortlist for the 2019 Ballon d'Or. On 3 November, Son was sent off during a 1–1 draw with Everton after sliding into André Gomes from behind, causing him to fall awkwardly and suffer a severe ankle injury. The injury prompted great concern and anguish from the players and supporters; Son was visibly highly distressed by the incident. Following a red card for a challenge on Gomes, Son also received suspension for three Premier League games. However, many professionals including former Everton player Kevin Kilbane expressed criticism of the red card decision, and Tottenham made an appeal to the Football Association against the dismissal. The FA accepted the appeal and Son's red card was rescinded on 5 November. Three days after this incident, in a 4–0 Champions League away game to Red Star Belgrade, in which Son scored a brace, rather than celebrating his first goal, he apologised to the camera for what happened at Goodison Park.
On 23 November 2019, Son scored Tottenham's first goal with José Mourinho as manager, earning him Man-of-the-match in a 3–2 Spurs victory. On 7 December, Son ran from one end of the pitch to the other, going past seven Burnley players, to score an individual goal that was labelled a goal of the season contender. The nature of the goal saw Mourinho dub him "Sonaldo Nazario" in reference to the type of goal former Brazilian international Ronaldo would score. On 22 December 2019, facing Chelsea, Son was sent off after raising his boot against Antonio Rüdiger's ribs. In January 2020, Son was awarded the Premier League goal of the month for December for his goal against Burnley, eventually winning him the Goal of the Season award as well. On 16 February 2020, Son scored two goals at Villa Park and earned a 3–2 victory for Tottenham, in which he became the first Asian footballer to score 50 goals in the Premier League, with 51 scored goals in 151 Premier League matches. Son played the full game despite suffering a fracture to his arm at the 31st second of the game. Head coach José Mourinho declared at a later stage that he was not optimistic regarding Son's injury and that it was likely that the South Korean would be sidelined for the rest of the season.