The Brazilian midfielder scored the only goal of the game just before the break when benefiting from a second bite of the cherry.
And it ended a run of six home games without loss in the league for the Clarets — who were edged out by Arsenal five months ago.
Burnley were the better side for much of the first half, but it was the visitors who went in ahead at the break.
Josh Brownhill tested Alisson with a right-footed piledriver from distance as the midfielder's search for a first Premier League goal continues.
Wout Weghorst, also on a mission to net his first goal for Burnley in the top flight, wasn't far away from opening the scoring, either.
The home side broke from a Liverpool corner through Ashley Westwood and the Dutchman only had the Reds' Brazilian stopper to beat from Jay Rodriguez's pass.
However, Trent Alexander-Arnold recovered to the goal-line and cleared as the former VfL Wolfsburg forward clipped the ball over the goalkeeper.
Alisson came to the away side's rescue again when Maxwel Cornet's set-piece broke fortuitously for Rodriguez off skipper Ben Mee, but the striker failed to make it two-in-two as the ex-Roma keeper saved at his feet.
In fact, it took Jurgen Klopp's former champions half-an-hour to pose a genuine threat in the final third.
Africa Cup of Nations winner Sadio Mane forced the block from Nick Pope when getting on the end of Alexander-Arnold's set-piece before the Senegalese forward steered the rebound wide.
Weghorst had another golden opportunity to break his duck in the moments that followed. The 29-year-old split Virgil van Dijk and Joel Matip from a Connor Roberts pass, only to smash his effort wide of the upright.
But if you give sides like Liverpool an inch they'll make you pay. And that's exactly what happened five minutes before Martin Atkinson's half-time whistle.
Burnley were flat-footed in their attempts to defend an Alexander-Arnold corner; Mane peeled away to provide the flick on in acres of space and Fabinho converted from close range at the second attempt.
The Reds were happy to keep the ball in the second half, taking the sting out of the fixture as their opponents struggled to land a game-changing punch.
Mee guided a header wide when getting above Matip to connect with Cornet's corner while Alisson collected efforts from Brownhill and Weghorst.
At the other end, Mee almost turned Alexander-Arnold's delivery into his own net before James Tarkowski's incredible intervention denied substitute Diogo Jota what seemed a certain goal from Mohamed Salah's assist.
5. Ben Mee 6.5
Another strong defensive display, cutting out several threatening crosses right in front of Pope, though a second half clearance almost found his own net. Went close with a header when out-jumping Matip, but lost Fabinho from Alexander-Arnold's corner. Photo: Stu Forster
6. Aaron Lennon 6.5
Rarely tested Robertson with the ball at his feet, with limited opportunities to get a run at the Scotsman, but his defence double act with Roberts proved very effective. Put in some gut-busting lungs to get back in support of his full back and tracked back well to nick possession on occasions. Photo: Gareth Copley
7. Maxwel Cornet 6
Much quieter since returning from Africa Cup of Nations, though we're unlikely to see the best of him when he's pinned to the touchline. Effective when playing off the shoulder, capitalising on Liverpool's high line, worked hard, but final product often lacking. Photo: Paul Ellis
8. Josh Brownhill 7.5
A brilliant box-to-box display from the Burnley midfielder. Labelled 'terrific' when singled out for praise by Dyche post match. Worked hard to suffocate Henderson, Fabinho and Keita in midfield and tested Alisson with a first half piledriver. Photo: Andrew Powell/Liverpool FC