A tense encounter on Merseyside saw both the Reds and Stoke struggle during one of the most instantly forgettable halves of football in recent years.
Both sides upped the ante after the break, however, with the Potters coming close to taking the lead on several occasions.
But it was Liverpool who did break the deadlock - Rickie Lambert heading against the bar before Glen Johnson, of all people, swooped into the box to head home the rebound after diving towards the ball and injuring himself in the process.
And 36-year-old Carragher praised Brendan Rodgers' side for earning a 'vital' win, hailing goalkeeper Simon Mignolet in particular.
Big win for Lfc the improvement in performance will be gradual so results at the moment vital. Well done to Mignolet huge save that.— Jamie Carragher (@Carra23) November 29, 2014
Mignolet proved plenty of his doubters wrong with some wonderful reflex saves against Stoke, especially one right at the death from a rocket of a strike from Bojan Krkic.
The Belgian did show the same old weaknesses in terms of hesitance on the ball and in the air but this was one of his better performances of the season and, as Carragher points out, he deserves plenty of praise for Liverpool's clean sheet.