Wing Phil MacKenzie's 52nd-minute try and three penalties by fly-half Gordon Ross put Premiership basement club London Welsh on the verge of only their fifth league win this season.
It would have closed the gap to two points on Sale Sharks - 14-13 conquerors of Bath on Friday night - but Gloucester fly-half Burns had other ideas.
Although the England international missed four shots at goal, he booted five others to keep Gloucester's title play-off hopes alive hopes.
London Welsh, in contrast, are now five points behind Sale with four games left, although that could change early next week.
Welsh have appealed a five-point deduction for fielding an ineligible player - scrum-half Tyson Keats - in the Premiership this season, and the result of their appeal hearing is expected to be made public during the next 72 hours.
They need it to work in their favour, given that their remaining Premiership fixtures are away against Bath and London Irish, plus home appointments with Northampton and Worcester.
London Irish took another major step towards survival with a comfortable 26-6 victory over a disappointing Worcester at the Madejski Stadium.
With this win Irish have at present a 10-point cushion over bottom club London Welsh and a five-point lead over 11th-placed Sale, with Irish still having to play those two clubs at home in their remaining four fixtures.
The game was a disappointing one for the annual St Patrick's celebration which saw an attendance of 19,523.
The driving line-out of the Irish was the dominant factor with all their tries coming from this facet of the game.
Worcester seemed incapable of dealing with it to ensure their first away win in the Premiership was never a possibility.
The Warriors were outbattled in the loose and were very limited in attack with lively wing David Lemi never receiving a pass.
Northampton's forward dominance eventually told as Christian Day's last-gasp try saw them enhance their play-off aspirations and end Wasps' unbeaten home record with a 26-24 result at a snowy Adams Park.
Northampton bossed the close exchanges and a penalty try plus further scores from Lee Dickson and Dylan Hartley - as well as four points from Stephen Myler's boot - gave them a 19-16 half-time lead.
Wasps had twice been reduced to 14 men as Rhys Thomas and Tom Palmer went to the bin in the first half, but Tom Varndell's try, two penalties and a conversion from Stephen Jones, and a three-pointer from Elliot Daly kept them in it.
And centre Daly's wonderful finish down the left, after another Jones penalty, briefly put them on course for a remarkable victory.
But there was still time for Ashley Johnson to become the third Wasp to receive a yellow card, and another punishing series of Saints drives ended with Day getting the crucial fourth try at the death, with Myler's conversion sealing the win.
And four penalties from Toby Flood steered Leicester to second place, at least for 24 hours, as they moved to within a point of leaders Saracens with a 12-9 victory in a tryless contest at Exeter.
The Chiefs had to settle for a losing bonus point, after Gareth Steenson had kicked three first-half penalties, as they remain in eighth place - but crucially 12 points outside the top four.
After a two-week break from Premiership action for the RBS 6 Nations, Exeter went into this game in front of a sell-out crowd at Sandy Park on the back of victories over London Welsh and reigning champions Harlequins.