The only goal of the two legs came from James Milner, whose early penalty proved to be the difference.
Jurgen Klopp’s men had the better chances but they were nearly on their way out of the competition on away goals as Konstantinos Stafylidis’ set piece was whiskers away with minutes left on the clock.
The Premier League side did not reach top gear, but Klopp will be delighted to see his side win ahead of Sunday’s League Cup final against Manchester City.
The Reds could not score in 90 minutes in Germany, but it took them just four minutes to open the scoring on Merseyside.
After Dominik Kohr handled in the box after jumping for a header, the spot-kick was awarded and Milner coolly slotted home his second successive Europa League goal at Anfield.
Roberto Firmino forced Marwin Hitz into making a save at his near post after a good low cross from Alberto Moreno.
Former Bundesliga star Firmino nearly doubled his side’s advantage but his 20-yard drive was hit straight at the goalkeeper.
The visitors could have been back in the contest as Lucas Leiva – playing centre-back for the Reds – sold Simon Mignolet short with a backpass and Caiuby Francisco da Silva could not make it count as he rounded the goalkeeper before shooting into the side netting from an acute angle.
Coutinho showed sharp link-up play when he played a one-two with Emre Can in the box before crossing for Sturridge to send a first-time shot creeping past the near post, but Hitz was there once again.
The England international – making his fourth appearance since returning from injury – nearly scored four minutes into the second half as Jurgen Klopp’s high pressing game forced the defence into making a mistake. Coutinho won the ball back and found Sturridge, who beat Ragnar Klavan in the box before pulling a shot across goal and past the post.
Liverpool dominated possession and they had the best chances in front of the watching crowd at Anfield.
In a rare opportunity for the away side, Tobias Werner of Augsburg looked to be through on goal but Mignolet, to his credit, was quick off his line to make contact with the ball and clear danger.
Sturridge was taken off after 65 minutes with his return from injury and Sunday’s final in mind and Liverpool were lacking his threat up front as they failed to increase their lead.
Jordan Henderson nearly scored from point-blank range, but Hitz was ever-present again. The visitors nearly drew level late on when Stafylidis’ free-kick went just past Mignolet’s post, as Liverpool saw out the victory to book their place in the next round.