Conway joined the Leinster Academy at the beginning of the 2009–10 season. Conway signed his first professional contract in 2010 for Leinster after impressing with the Academy. He made his senior debut for Leinster in a Magners League fixture against Scarlets on 20 February 2010, coming on as a late substitution and setting up Jonathan Sexton's try. He made his starting debut for Leinster on 7 March 2010, with a victory over the Cardiff Blues. Conway made his Heineken Cup debut in Leinster's 2012–13 Heineken Cup pool fixture against Exeter Chiefs on 13 October 2012. Conway scored a hat-trick for Leinster on 3 May 2013, in their Pro12 game against Ospreys. He started for Leinster against Stade Français in the 2012–13 European Challenge Cup Final on 17 May 2013, which Leinster won 34–13. Conway also started for Leinster in the 2012–13 Pro12 Final victory against Ulster on 25 May 2013, which Leinster won 24–18.