The Clarets remain in the Premier League’s bottom three but showed the stomach for the fight as they pushed title-chasing Liverpool all the way in a rousing Turf Moor performance.
Jack Cork’s second league goal of the season nine minutes into the second half threatened to end the Reds’ unbeaten record as Burnley sensed an end to their seven-game winless run.
Two goals in seven minutes from the ageless James Milner and substitute Roberto Firminho dashed those hopes as Liverpool’s rapier-like response brought them back to within two points of defending champions Manchester City at the top of the table.
But Sean Dyche’s new-look side showed the character and ability that remains within the camp as they restored belief with a performance they will need to repeat and which wasn’t reflected in the margin of a Liverpool win embellished by Xherdan Shaqiri’s injury-time strike.
Dyche made five changes to the Clarets’ starting line-up in a bid to shake things up following the dismal showing at Crystal Palace at the weekend.
And the response was just what the manager – and the home supporters – would have wanted as Burnley took the initiative to rattle the unbeaten Reds from the off.
Phil Bardsley, back in the side at right-back, set the tone with a couple of thunderous early tackles which immediately lifted the mood inside Turf Moor.
He was also responsible for creating the first chance of the night with an early ball forward to the recalled Ashley Barnes.
Barnes helped it on to Chris Wood, but the striker delayed in getting his shot off as the opportunity passed him by.
Bardsley teed up Wood again with a booming diagonal pass for a shot which was deflected over the bar.
And from the corner that followed, Bardsley wasn’t far away with a rasping long-range drive which flew just wide of Allison’s right-hand post.
Liverpool boss Jurgen Klopp also shuffled his pack, making seven changes to the side that had won Sunday’s Merseyside derby with Everton.
But his re-shaped side was never allowed to find their feet as Burnley pressed all over the pitch with the kind of up-tempo performance that had gone missing at time this season.
The visitors were given few sights of Joe Hart’s goal, Daniel Sturridge wasting their best opening with a poor touch from Milner’s cross moments after throwing himself to the floor in disappointing fashion to secure a free-kick.
And momentarily it was the Clarets who thought they had grabbed the opening goal to lead at the break, Barnes superbly volleying home a wickedly delivered free-kick from Robbie Brady before having his celebrations curtailed by an assistant referee’s offside flag.
The pace didn’t drop at the start of the second half, although this time it was Liverpool doing the pressing.
And it took two fine saves from Hart to keep the visitors at bay.
The first, turning away a Sturridge shot, was more regulation for the England international.
But the second was right out the top drawer as he flew across his goal to finger-tip a Naby Keita shot wide via the outside of the post.
And Burnley capitalised in rousing style to go in front.
Liverpool boasted the best defensive record in the division and were one clean sheet away from setting a new top-flight record for the fewest goals conceded at this stage of a top-flight season.
But they were breached for only the sixth time as the Clarets’ hunger paid off following a corner from Johann Berg Gudmundsson.
Allison was only able to parry James Tarkowski’s header and when Barnes beat him to the punch to the loose ball, Cork was on hand to tap into an empty net.
Liverpool’s response was to warm up Firminho and top scorer Mo Salah.
But before they made it on the Reds were level as Burnley’s advantage lasted just five minutes, Milner picking up a lay-off from Origi to fire through a crowd of bodies past an un-sighted Hart and into the bottom corner.
And when Firminho did join the action, he scored with his first touch as Liverpool went ahead to complete the turn-around.
Trent Alexander-Arnold’s floated pass from a short free-kick unlocked the Clarets’ defence and when Virgil van Dijk got around the ball to steer the ball across the face of goal, Firminho couldn’t miss from two yards out.
Ashley Westwood, another to bring some added energy to the claret and blue ranks, shot at Allison as Burnley tried to respond.
But despite the heroics of Tarkowski and Bardsley at the other end, as they followed up a double save from Hart with incredible blocks to deny Keita, the Clarets couldn’t bridge the one-goal gap and gain the reward their performance deserved as Ben Mee’s last-minute header was clawed away by Allison.
The goalkeeper then looked to have carried the ball out of play at the expense of another corner, but was allowed to launch the counter-attack which ended with Shaqiri’s clinical finish as Liverpool bagged the points which makes it their best ever start to a season after the first 15 games.
Burnley: Hart, Bardsley, Tarkowski, Mee, Taylor; Gudmundsson, Westwood, Cork, Brady (Lennon 71); Barnes (Vydra 84), Wood (Vokes 71). Substitutes: Heaton, Lowton, Hendrick, Long.
Liverpool: Alisson; Gomez (Alexander-Arnold 23), Matip, van Dijk, Moreno (Salah 65); Keita, Henderson, Milner; Shaqiri, Sturridge, Origi (Firminho 66). Substitutes: Mignolet, Fabinho, Lallana, Camacho.
Referee: Stuart Attwell