The hosts initially coped better with the awful Lancashire conditions, creating more chances than Liverpool, only for their wastefulness to prove costly as midfielder Fabinho bundled in what proved to be the winner five minutes before the break.
Chasing a goal that would have earned Burnley their fourth successive draw, they pumped plenty of long balls forward and crosses into the box in the second half, but failed to create a clear-cut chance.
Liverpool dug in defensively, with goalkeeper Alisson standing firm when called upon, to see out a sixth successive win in all competitions.
Victory moved Liverpool on to 54 points from 24 matches and if they win their game in hand on City the gap at the top would be just six points.
Burnley will see this as a missed opportunity and remain rooted to the bottom of the table, seven points from safety, but having played two fewer games than Newcastle United in 17th.
“It is always hard to play here,” Fabinho said. “Burnley is a tough place to come. Sometimes we forget to play football a little bit because we want to fight for the second balls but when we had the ball on the ground we created chances.
“We didn’t score the second goal but defensively the team was very good today. A nice three points.”
Liverpool forwards Mohamed Salah and Sadio Mane made their first starts since returning from the Africa Cup of Nations in the driving rain, but could only look on as Burnley started on the front foot.
The first effort at goal was from Burnley after Josh Brownhill’s fierce strike was tipped wide by goalkeeper Alisson at full stretch.
It took Liverpool until the 21st minute to test home stopper Nick Pope, with Naby Keita denied low to the keeper’s right before, down the other end, Burnley’s January signing Wout Weghorst should have done better when through on goal.
The home side remained on top and soon forced Alisson into another save, the Brazilian smothering at the feet of Jay Rodriguez.
The chances were coming thick and fast, with Mane then immediately firing straight at Pope from a free kick, before Weghorst should have hit the target from a good position for Burnley.
Burnley were left to rue their profligacy when Mane flicked on Trent Alexander-Arnold’s corner to Fabinho, who squeezed the ball over the line at the second attempt – the Brazilian’s fifth goal in all competitions since the turn of the year.
“You have to take your chances,” Burnley coach Sean Dyche said. “Two or three of the chances we created were huge chances.
“There are a lot of good things coming out of these players, but we got nothing for it. We have to keep playing like that against all standards of opposition.”
It was a matter of just holding on for Klopp’s side given the conditions and determination of Burnley after the break.
Alisson made more saves in a single league match than he had all season, much to the frustration of a soaked Turf Moor crowd.
Substitute Diogo Jota should have added a late second but fired wide with goal at his mercy. It mattered little, as Liverpool celebrated a hard-earned win at full time.
“I am delighted with the way we played in the circumstances,” Klopp said. “You can only feel the wind like that if you were on the pitch. It came from all directions.
“We always deal with the circumstances. We defended many situations really well – I am really happy.”