Osbourne made his Hibernian début on 14 August 2011 in a 4–1 away loss at Kilmarnock, coming off the bench at half-time to replace Lewis Stevenson. Since joining the club, he was involved in the first team in a number of matches, playing in the midfield position. On 24 September 2011, Osbourne scored with a deflected shot in a 3–3 home draw with Dundee United, but it was officially recorded as a Scott Robertson own goal. However, he suffered two separate injuries and was sidelined for weeks. Osbourne made his return to the starting line–up against Dunfermline Athletic on 14 January 2012, starting a match and playing 86 minutes before being substituted, in a 3–2 win. Following his return from injury, he continued to regain his first team place for the rest of the 2011–12 season, playing in the midfield position. Osbourne finally scored his first goal for Hibernian, scoring in the 18th minute, in a 4–3 loss against Motherwell on 22 February 2012. Having aimed to help the club avoid relegation, he successfully predicted with his target after helping Hibernian beat Dunfermline Athletic 4–0 on 7 May 2012. However, Osbourne started in the Scottish Cup Final against rivals, Hearts, as the club lost 5–1. Despite missing two more matches halfway through, he scored one goal for Hibs in 35 appearances during the 2011–12 season.