The hosts went ahead in the ninth minute as a corner found its way to Lees-Melou, whose shot from the edge of the penalty area took a deflection off Josh Brownhill and flew in after clipping the post.
Pukki fired into the far corner in the 86th minute to double the lead after latching on to Mathias Normann's superb through ball which caught Burnley's defence totally off guard.
Earlier, Burnley's Maxwel Cornet sliced a low cross by Dwight McNeil wide from point-blank range and Norwich wasted opportunities to put the game to bed, as Grant Hanley headed onto the woodwork and Milot Rashica had a shot saved.
Norwich's first league victory since January took them to 21 points with seven games remaining and followed last weekend's 0-0 draw at Brighton & Hove Albion, giving manager Dean Smith hope they can pull off a miracle and escape the drop.
"It [the result] shows we're still fighting, that was important for our supporters. It gives our dressing room an awful lot of belief," Smith told Sky Sports.
"We had hoped results would have gone a little better for us yesterday but we can only concentrate on ourselves... if you win two or three games on the spin, the world looks different."
Smith said he expected more resolute defending when Norwich travel to Manchester United for their next match.
"It's a tough place to go there and get a result. I got a win there with Aston Villa so hopefully I can do the double."
Defeat was especially disappointing for Sean Dyche's side who remained 18th on 24 points from 30 games, four points below Everton and nine adrift of Leeds United after both those teams won on Saturday to boost their survival bids.
"There's still eight games left and we've won games that have been written off," Brownhill, whose side visit West Ham United next, said.
"We've been here before. We're a fighting team but we've not managed to be on the right side of the result."