Divock Origi missed a golden chance for Liverpool just before half-time as both sides went level into the break, with the Reds continuing to be left frustrated as in recent weeks.
Liverpool played into the visitors' hands in the second half, with Sean Dyche's side executing their gameplan and topping things off with an Ashley Barnes penalty to earn a 1-0 win.
It's the champions' first Premier League defeat at Anfield in nearly four years, with Jurgen Klopp's side now six points behind league-leaders Manchester United and with no signs of improving their barren run.
Liverpool had a flurry of early corners and came close on two occasions through Fabinho and Sadio Mane, but the ball failed to find its way into the back of the net - an unfortunately familiar sight from recent weeks.
The rhythm of the game was clear from the off as Burnley kept their defensive shape well but looked to break quickly on the counter with long and direct balls.
Just seconds later, Alisson made an excellent save from Ashley Barnes' strike at the edge of the box after initially being beaten to the ball by Chris Wood.
The Reds came inches away from taking the lead as Shaqiri drilled a powerful strike just wide of the post after finding space at the edge of the box.
Oxlade-Chamberlain had a pop at goal from distance a couple of minutes later, before Origi joined in with another low strike which was dealt with by goalkeeper Nick Pope.
The Belgian had another strike after a quick counter-attack led by Mane, but the shot was straightforward again for Pope, who was behind the ball the entire way.
But Origi should have made it 1-0 just before half-time after Ben Mee's mistake saw him go one-on-one with the goalkeeper. Origi went for an early finish but his strike agonisingly hit the woodwork as the Reds' fortunes in front of goal showed no signs of improving.
Fabinho was shown a yellow card for a coming together as the half-time whistle blew, with both managers and sets of players getting into a scuffle before eventually ushering back into the tunnel.
Pope was called into action early in the second half as Alexander-Arnold had a low drive at goal, with the Clarets keeper doing well to get down low.
Klopp had seen enough before the hour mark as Salah and Firmino both came on to the pitch, replacing Origi and Oxlade-Chamberlain.
A fantastic run and burst of energy by Wijnaldum saw him set up Salah, who went for the near post and saw his shot saved by Pope - who seemed intent on repeating his heroics from last season's visit of Anfield.
Next, Alexander-Arnold won the ball high up the pitch before finding Salah, who ran into the box and centred for Mane, who blazed over the crossbar whilst under pressure from a Burnley defender.
Another lapse by Burnley whilst playing out at the back saw Robertson set up Firmino, who disappointingly skewed his shot away from goal.
With less than ten minutes left, Alisson was harshly adjudged to have brought down Barnes inside the box, with the Clarets striker taking the ensuing penalty and dispatching it past the Reds goalkeeper.
The Reds had it all to do as Takumi Minamino came on for Shaqiri, but it wasn't to be as their title defence dealt a further blow with a disappointing 1-0 defeat.
Liverpool XI: Alisson, Alexander-Arnold, Fabinho, Matip, Robertson, Wijnaldum, Thiago, Shaqiri, Oxlade-Chamberlain, Origi, Mane.
LFC Globe's MOTM: Nick Pope
Next Fixture: Man United (FA Cup, A) - 24 Jan, 5pm