Having played out a goalless draw in the first leg, Juergen Klopp's side spurned several chances after James Milner's early spot kick and survived a nervy finale to book a place in the next round.
There was no such tension for Gary Neville's Valencia whose recent upturn in form continued with a 4-0 win at Rapid Vienna to complete a 10-0 aggregate success, while Shakhtar Donetsk, Athletic Bilbao, Lazio and Bayer Leverkusen also progressed.
Shakhtar completed a superb 3-0 win at Schalke 04 to progress from the round of 32, while Bilbao's Sabin Merino scored an 81st-minute equaliser as they drew 1-1 at home to Olympique de Marseille and went through 2-1 on aggregate.
Lazio beat Galatasaray 3-1 in Italy with all four goals coming in 14 second-half minutes as Marco Parolo, Felipe Anderson and Miroslav Klose netted for the Serie A side, who went through 4-2 over the two legs.
Bayer Leverkusen's Karim Bellarabi again proved decisive in their tie with Sporting Lisbon as the Germany international, who scored the winner in the first leg, netted two more in a 3-1 win that secured a 4-1 aggregate success.
Original source: Liverpool through with nervy win