The Reds would have wanted to put this tie to bed as quickly as possible with the Wembley showpiece against City on Sunday and they were aided by the early penalty.
Liverpool pressed high up the pitch and played some stylish football but, with six goals in seven previous Europa League outings this season, they again struggled in front of goal and failed to turn their superiority into a bigger lead.
Brazil midfielder Philippe Coutinho had a promising chance when he was set up by Daniel Sturridge, only to see his stabbed shot from eight yards saved by keeper Marwin Hitz.
England international Sturridge had a shot palmed on to a post after a Coutinho pull-back and the striker also sent a shot just wide when he was not flagged after appearing offside.
The anxiety for the home side increased as the game went on – and Reds captain Jordan Henderson had a close-range shot saved before the relief of the final whistle.