Burnley got off to a fine start when Ashley Barnes prodded in Matt Lowton's fine pass with seven minutes gone.
Liverpool struggled to find their momentum but were level on the stroke of half-time thanks to Wijnaldum, who converted Divock Origi’s cross for the equaliser.
The winner came from Can's low drive from distance on 61 minutes
Liverpool are still fourth, but the victory takes them to within a point of second-placed Tottenham Hotspur and five clear of Arsenal, in fifth.
Burnley remain 12th on 31 points.