Edouard was born in Kourou, French Guiana. He joined the Paris Saint-Germain Academy in 2011 from amateur side AF Bobigny. His first noticeable achievement was during the 2013–14 season when he was the club’s top goalscorer at U17 level; scoring 25 goals in 22 appearances in the U17 National Championship. He scored 22 goals in 14 league appearances for the U17s the following season and was part of the squad which won the Al Kass Cup, ending the competition with a return of three goals in five appearances. He also represented the club’s U19, playing two UEFA Youth League matches, a Coupe Gambardella match and seven U19 Championship matches during which he scored seven goals. In two years, he scored more than 60 goals for the club at youth level, earning him the nicknames of “The Rocket” and “Magic Odsonne”.

During the 2015–16 season, Edouard played with both the U19 team and the CFA team. In January 2016, he won the 2015 Titi d’Or, an award voted for by the club’s supporters for the best academy player for a particular season. The award was notably won by players such as Kingsley Coman and Jean-Kévin Augustin in previous years. PSG also made it to the Youth League final but were ultimately beaten 2–1 by Chelsea. Edouard scored three goals and registered three assists during the competition.

On 27 April 2016, he signed a professional contract with Paris Saint-Germain’s senior side. During the 2016–17 pre-season, Edouard was included in PSG’s squad for the 2016 International Champions Cup. Having previously featured as a substitute, he came on in the 79th minute against Leicester and scored in his non-competitive debut, making it 4–0 in the closing minutes of the club’s last International Champions Cup game. As a result of the victory, PSG were crowned International Champions Cup champions.

On 8 August 2016, Edouard joined Toulouse on a season-long loan. He made his debut for the club on 14 August 2016 against Marseille, replacing Issiaga Sylla after 74 minutes in a 0–0 away draw. Edouard scored his first professional goal on 19 November against Metz, scoring a consolation goal late in added time in a 2–1 home loss.

On 30 March 2017, Edouard was suspected of having shot at a passer-by with an airsoft gun on 11 February, injuring him in the head. As a result of the incident, he was interrogated by police and his loan was prematurely terminated by Toulouse. At the time of his loan being cancelled, Edouard had made 17 appearances and scored one goal for the club. It was later revealed that his teammate, Mathieu Cafaro, had confessed to being the one who had fired the gun. Cafaro later recanted, however, and on 13 June the Toulouse Prosecutor asked that Edouard be charged with a four months suspended prison sentence and a €6,000 fine for his involvement in the incident. Edouard alleged that Cafaro fired the shot while Cafaro claimed not to have been in the car at the time of the shooting. On 4 July, Edouard was sentenced to a suspended four months’ prison sentence by the Correctional Court of Toulouse. He was also ordered to pay a fine of €6,000 as well as damages of €2,600 to the victim for the wounds suffered to his ear.

Edouard signed for Scottish Premiership club Celtic on 31 August 2017 on a season-long loan deal. He scored on his debut away to Hamilton Academical on 8 September, helping Celtic to a 4–1 victory. Later that year, on 2 December, he scored his first career hat-trick in a 5–1 win over Motherwell at Celtic Park. He made his UEFA Champions League debut three days later, coming on as a substitute for compatriot Moussa Dembélé in a group stage defeat to Anderlecht.

On 11 March 2018, he scored the winning goal in the 69th minute of a 3–2 win over Old Firm rivals Rangers at Ibrox Stadium, and on 29 April he added two more goals in a 5–0 win over the same opposition to seal his club’s seventh consecutive league title. He ultimately made 29 appearances for the season and scored 11 times as Celtic completed a domestic treble.

On 15 June 2018, Edouard signed a four-year deal with Celtic for a fee that the club said was the highest in their history. The amount would have had to have exceeded the £6 million that the club paid for Chris Sutton and John Hartson in 2000 and 2001, respectively, with sources estimating it as over £8 million. The following month he was named on the 100-man shortlist for the 2018 Golden Boy award; he was the only footballer playing in Scotland at the time to be nominated for the accolade. On his second debut for the club, Edouard scored the opening goal in a 3–0 win over Alashkert in the first qualifying round of the 2018–19 UEFA Champions League. The Frenchman continued his scoring run in the following round, netting twice in a 3–1 victory against Rosenborg at Celtic Park with manager Brendan Rodgers describing him as a ‘top striker’ after his displays.

On 25 May 2019, Edouard scored twice as Celtic beat Hearts to secure the Scottish Cup and a historic “treble treble”, that is, winning the Scottish Premiership, the Scottish League Cup, and the Scottish Cup in three consecutive years.