Emre Can struck just after the hour mark to secure victory for Liverpool, although it was a far from convincing performance by Liverpool.
Liverpool: Emre Can's superb shot guided Liverpool to a 2-1 home win over Burnley on Sunday and maintained their bid to qualify for next season's Champions League.
Ashley Barnes, with his club's first goal at Anfield since 1975, had given Burnley a deserved early lead only for Georginio Wijnaldum to equalise in first-half injury-time.
And Can struck just after the hour mark to secure victory for Liverpool, although it was a far from convincing performance by Jurgen Klopp's side.
German international Can collected a short pass from Divock Origi and was allowed space to unleash a right-foot shot from 25 yards which the Burnley goalkeeper, Tom Heaton, might have been expected to save before it reached the bottom left-hand corner of his net.
Victory saw Liverpool remain fourth, still in the final Champions League place open to English clubs.
The win moved them to within a point of third-placed Manchester City but, more significantly, saw Liverpool go five points clear of fifth-placed Arsenal.
"The performance was good enough to win the game but the result is massive for us, it's the first ugly game we have won," Liverpool manager Klopp told Sky Sports. "Usually when we are not that good we lose."
Meanwhile Burnley boss Sean Dyche was left bemoaning his side's lack of good fortune.
"It's a tough one to take, because everyone gave a really good account of ourselves and we went up with a sublime goal," he told the BBC. "We just needed a scratch of luck along the way."
Liverpool had gone in at the half-time interval level although it was a scoreline that even the most ardent home supporter would have admitted was harsh on Burnley.
With just two away points to their name all season -- from a still impressive tally of 31 -- Dyche's team opened the game aggressively and took the lead in just the seventh minute.
Right-back Matt Lowton crossed low into the area from the wing and, after Andre Gray narrowly missed making contact, Barnes slid in ahead of his marker to force the ball past Liverpool goalkeeper Simon Mignolet.
Burnley, without a victory at Anfield since 1974, would have hoped to reach the interval without conceding an equaliser and looked well on course for that target for the majority of the half.
Not until the 45th minute did Liverpool enjoy a 'shot' on target and that was actually an over-hit defensive header from Burnley's Joey Barton which forced Heaton into a comfortable diving save, his first of the game.
But a minute into first-half injury-time, Liverpool claimed an unlikely equaliser which raised the mood around the stadium.
Origi, starting in place of the injured Roberto Firmino, crossed from the left and Wijnaldum made a lazy attempt to control the ball, flicking out a boot and benefitting from a fortunate rebound from defender Ben Mee, before drilling home an unstoppable finish from six yards.
Even before the goal, Burnley's crosses from the right had caused concern, with Gray's chip almost finding Barnes after five minutes.
Solid at the back, Burnley also looked the more likely to score a second goal after Barnes's opener.
But the second half opened with Liverpool in far more determined mood, although their suspect defending always gave Burnley hope.
Barnes sent a speculative first-time shot wide soon after the restart before Liverpool finally threatened, with Joel Matip's shot from a Philippe Coutinho cross blocked
Can's goal was still harsh on Burnley, although the visitors almost responded just three minutes later, when Mee headed a free-kick across goal and Barnes's effort was blocked by Ragnar Klavan.
Both teams squandered late chances for further goals. Sadio Mane had only Heaton to beat but saw the Burnley keeper tip his 12-yard shot over the bar in the closing stages before Lowton missed a glorious opportunity to equalise, shooting wide from a scramble in the area in stoppage-time.