Liverpool avoided potential embarrassment in Romania as they came from behind after Bruno Fernandes' header gave Unirea Urziceni the lead.
Goals from Javier Mascherano from long range and Ryan Babel from close in turned things around before the interval and Steven Gerrard's 33rd European goal wrapped up a 4-1 aggregate success for Liverpool.
Liverpool now face French side Lille (25/1) next while Fulham face Italian giants Juventus next.
Europa League Outright Winner OddsLiverpool 6.50 at Bet365
Juventus 8.00 at Interwetten
Valenica 9.00 at Bet365
Benfica 12.00 at Betsson
Hamburg 12.00 at Betsson
Werder Bremen 16.00 at Interwetten
Atletico Madrid 16.00 at Interwetten
Rubin Kazan 17.00 at Bet365
Wolfsburg 17.00 at Bet365
Panathinaikos 21.00 at PaddyPower
Fulham 21.00 at Bet365