Sean Dyche's side had taken a shock lead with less than 10 minutes on the clock, Ashley Barnes opening the scoring. However, Georginio Wijnaldum levelled things up on the stroke of half-time, before Can's stunning strike put the Reds ahead.
Burnley pushed hard for an equaliser late on but couldn't find a goal, Liverpool gaining valuable points in their quest for a Champions League place.
Boyd and Barnes combine for opener
Having troubled Liverpool with their direct style in the opening minutes, Burnley took a lead with a goal that seemed crafted out of a Pep Guardiola footballing manual, rather than Sean Dyche's.
Picking the ball up in midfield, George Boyd moved forward and split Liverpool's defenders with an inch-perfect pass, finding the run of Ashley Barnes at the back post, Barnes finding the net with an incisive finish past Simon Mignolet.
Upset by Burnley's pace and power, and a string of fouls given by the referee, Liverpool failed to test Tom Heaton for the opening half an hour. Burnley weren't exactly banging at the door for a second, but Andre Gray did enjoy some space on the counter, sending a shot wide of Mignolet's goal from a tight angle.
Wijnaldum equaliser welcomed by crowd
Just as they seemed to be running out of time before the break, the equaliser finally came for the hosts. Philippe Coutinho's quick-feet in midfield released Divock Origi down the left, with Origi's cross falling fortunately at the feet of Wijnaldum. Pausing for an extra second to send Heaton the wrong way, the Dutchman applied a composed finish to level things up.
Desperate to take their momentum into the second-half, Liverpool were out quickly but met Burnley's brick wall as the game got back underway, Dyche seemingly able to lift his players after that late goal.
Coutinho was struggling to make an impact and after one failed pass too many, Klopp opted to replace him on the hour mark, taking his £150,000 a week Brazilian off for 17-year-old Ben Woodburn.
Can-tastic strike to win it
Although Woodburn wasn't involved, Klopp was delighted to see his side complete their comeback just seconds later. Emre Can, who too had suffered a poor game up to that point, showed his continued confidence by lining one up from 25-yards, sending the ball curling into the bottom right hand corner, Heaton well beaten.
Again Burnley refused to let their heads drop, and they'd have been level soon after if not for the fantastic intervention of Ragnar Klavan, who put his body on the line to block Ben Mee's shot from the edge of the area.
Ben Woodburn had been brought on to help Liverpool find a goal, but with his side now leading chances for creativity were limited, a nice cross just missed by Origi and Mane his best contribution in the next 15 minutes or so.
Taking Gray off for Sam Vokes, Dyche urged his team to go even more direct as the game entered the final stages, Mignolet coming good with a number of fine punches to keep the Clarets at bay. A half-chance for Barnes to get his second came about as the ball dropped to him for a half-volley, one where he blazed over.
Liverpool would have expected to have chances to bag a third on the counter, but in truth they struggled to keep hold of the ball for a meaningful amount of time.
On chance did come for Mane in the final minute of normal time, Can finding the Senegalse winger with an intricate dink over the defence, only for Tom Heaton to push his shot over.
Burnley's best chance to equalise came deep into stoppage time. A familiar sight for Liverpool fans, they failed to win the second-ball after a set-piece, Lowton looking to take advantage as the ball bounced to him just yards out, the away fans looking on in horror as he shot over.