The England forward had endured a 22-game goal drought for club and country until he set Liverpool on their way to a 3-1 away win with a close-range header.
Lazar Markovic grabbed the second before a trademark Sterling run ended with his fifth of the season to set up a semi-final meeting with Chelsea.
Adam Lallana played 90 minutes for the Reds while England U20s defender Baily Cargill featured for Bournemouth.
England Under-21s foward Harry Kane scored again as Tottenham Hotspur secured a final-four showdown with Sheffield United with a 4-0 victory over Newcastle United at White Hart Lane.
Kane was joined in the Tottenham starting line-up by Three Lions squad players Danny Rose and Andros Townsend, while England U18s hitman Adam Armstrong played half an hour for Newcastle.
