The Reds made the breakthrough in the fifth minute after midfielder James Milner slotted home from the penalty spot.
Liverpool had 24 attempts on Augsburg goal of which six were on target but the likes of Philippe Coutinho and Daniel Sturridge couldn’t extend the home side’s advantage.
However, Jurgen Klopp’s side managed to dig deep and hold on despite late pressure from the Bundesliga side and seal a 1-0 aggregate win to book their place in the Europa League last-16 draw.
Former Liverpool goalkeeper James felt three successive clean sheets for the Reds was a big positive ahead of Sunday’s League Cup final.
“The thing I like is that they’re keeping clean sheets. Three in a row now in all competitions. It is a steady side for three games in a row,” James told BT Sport.
“Of course, we all want to see better chances created. They opened up the game quite often but the choice of finishing was slightly off.
“We want to see Liverpool win 5-0 because Augsburg aren’t a great side.”
Liverpool are bidding to win Europe’s second tier competition for the first time since 2001.
The Reds reached the Europa League semi-finals in 2010 but Diego Forlan’s late winner denied Rafael Benitez’s side a place in the final.
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