Liverpool did not have it all their own way in the first half, though, with Ashley Barnes opening the scoring for the visitors as they eyed a maiden away win this term and a first seasonal double over the Reds since 1930.
But Georginio Wijnaldum equalised in first-half stoppage time and Can's long-range stunner completed the turnaround just past the hour, clinching the points with Burnley's attempts at a leveller of their own coming up short.
Klopp's men took until the sixth game of the calendar year to taste a league victory in 2017, but they have now won three of their last four with a trip to City's Etihad Stadium to come next Sunday.
It had looked as though the Reds' woes against middling opposition had returned when Burnley deservedly led after seven minutes.
Matthew Lowton delivered a superb low cross from the right and Barnes, arriving behind Joel Matip at the back post, stretched to finish high past Simon Mignolet.
The visitors retained a threat on the break through Andre Gray and he shot across the face of goal from a narrow angle, but, having previously offered little in response, Liverpool drew level on the stroke of half-time.
Divock Origi drifted wide to the left and cut back inside to deliver a cross towards Wijnaldum, who saw his attempted flick blocked but then pounced on the rebound to wrong-foot Tom Heaton with a calm strike.
1 - Georginio Wijnaldum's equaliser came through Liverpool's first shot on target of the game. Patience.— OptaJoe (@OptaJoe) March 12, 2017
Barnes sent an effort dipping wide shortly after the restart, but Liverpool finally took control with just under half an hour remaining.
Origi was again the provider as he squared for Can, 25 yards from goal, to step forward and drill beyond Heaton into the bottom-right corner.
The hosts were given an immediate reminder of Burnley's menace as a ball dropped for Barnes in the area from a set-piece, but Ragnar Klavan dived across to block magnificently.
Ben Woodburn, on for Philippe Coutinho, flashed a cross just past the right-hand post and only a fine Heaton save kept Sadio Mane from adding a third, before Burnley's best late chance fell to right-back Lowton, rather than a striker, and he blazed over.
KEY OPTA STATS
- Liverpool have now won 16 Premier League games this season; equalling their tally of wins for the entire 2015-16 season.
- Burnley have taken just two points from a possible 42 away from home in the league this season (W0 D2 L12).
- Liverpool have won 14 points from losing positions in the Premier League this term; a joint-high with Tottenham.
- The Clarets had just one shot on target in the game – no team has had one or fewer shots on target in an away game more often than Burnley in the top-flight this season (4).
- Ashley Barnes (10) is now just one goal short of tying with Danny Ings (11) for the most Premier League goals for Burnley.
- Liverpool conceded inside the first 10 minutes of a Premier League game at Anfield for the first time since August 2015 (v West Ham, Manuel Lanzini).
- All 16 of Georginio Wijnaldum’s Premier League goals have been scored in home matches (five for Liverpool, 11 for Newcastle).
- Since Jurgen Klopp’s first game in charge, the Reds have scored more Premier League goals from outside the box than any other team (21).