Jordan Henderson struck in the first half to help Liverpool leapfrog Group A leaders Anzhi Makhachkala who lost 3-1 at Young Boys in Berne.
All three sides finished with 10 points but Anzhi's better record over the Swiss club in a three-way head-to-head took them into the knockout stages.
Former European champions Steaua's 10 men held Copenhagen 1-1 to progress as Group E winners on 11 points with Stuttgart in second on eight and level with the Danes but the Germans progressed owing to a better head-to-head record.
The last two positions will be filled later on Thursday, with Basel and Hungary's Fehervar vying for the remaining berth in Group G. Premier League Tottenham Hotspur take on Panathinaikos for the final Group J slot. (Writing by Tom Pilcher, Editeing by Ed Osmond)