Healy's chance arrived ten minutes into the second half when Michael Duff tripped winger Eddie Lewis just inside the left corner of the penalty box and he drilled his spot-kick into the bottom right corner of the net
Then, with 20 minutes left, Blake curled a superb 20-yard free-kick into the top corner for his fourth goal of the season.
Matthew Kilgallon headed against Burnley's bar from six yards after only five minutes and Lewis shot into the sidenetting, but Leeds were on the back foot for much of the first half as they struggled to contain the darting runs of Nathan Dyer who filled the gap between the midfield and Ade Akinbiyi in the attack.
Neil Sullivan needed an acrobatic save to keep out James O'Connor's shot from distance, Dyer's deflected effort from just outside the box cleared the bar by a fraction and John McGreal headed narrowly wide as the Clarets threw men forward.
After their well publicised spat in the tunnel at Turf Moor last month, Leeds' Shaun Derry and Burnley's Micah Hyde were inevitably jeered by the rivals fans and it was a particularly strong tackle by Derry on Hyde in the build-up to Lewis' near miss.
Derry, who was clearly fired up for the contest, also covered yards of ground to complete a brave sliding tackle on Dyer.
Rob Hulse had the ball in Burnley's net three minutes after the break, only to have his celebrations cut short by a linesman's flag and visiting defender Jon Harley also had a 58th minute goal disallowed for taking a free-kick too quickly.
A crowd of 21,318 saw Leeds consolidate their place in the top four.