The visitors appeared in control as they opened up an 18-9 interval lead thanks to tries from Shane Monahan and Tom Savage, with Burns involved in the creation of both scores.
But the boot of Gordon Ross, who ended the game with 20 points to his name, kept Welsh in it and Franck Montanella's try looked to have set Lyn Jones' men up for a third-straight win.
However, a thrilling counter-attack instigated and supported by Burns, back after a spell in the sin bin, saw replacement prop Shaun Knight barrel over for the clinching score.
Burns and Ross had traded early penalties before the visitors landed the first telling blow.
Welsh had lost hooker Neil Briggs and centre Sonny Parker to injury when Burns struck his second penalty before producing two moments of brilliance to create the opening try.
The fly-half launched a break from deep with a lovely sidestep and only Nick Scott's deliberate batting down of the scoring pass prevented a try.
The Exiles wing was sent to the sin bin for his trouble and Burns caught the hosts sleeping as he took a quick tap to allow the powerful Monahan to get over on the right.
Ross responded with a penalty for Welsh but they were fortunate not to concede a second try to Monahan when the winger brilliantly swerved his way out of the grasps of the Exiles cover, only to be denied a memorable score by a superb covering tackle from Tyson Keats.
It proved a brief reprieve as more good work from Monahan and wing partner James Simpson-Daniel laid the platform for Savage to barge over after Burns had gone close.
Ross again responded with a penalty but Welsh were being forced into ill-discipline as they struggled to contain Gloucester's desire to put tempo on the game.
They were also guilty of failing to convert two excellent tryscoring chances. Lock Matt Corker sparked a thrilling counter from deep during the dying embers of the first half, but repeated battering at the Gloucester line yielded no reward.
And moments into the second half Ross failed to get his pass away with several men free outside him as Burns stepped in and was adjudged to have deliberately knocked on.
That saw the Gloucester man binned and Ross kicked his fourth penalty, but Welsh infringed moments later after failing to secure their own line-out ball to let Billy Twelvetrees reinstate the nine-point gap between the sides.
When the Exiles did manage to win set-piece ball their rolling maul was proving a handful for Gloucester, and one such surge led to Ross' fifth penalty.
The hosts then took the lead after more strong work from their pack. Several drives took them close to the line before Montanella was adjudged to have grounded the ball beneath a pile of bodies.
Ross converted for a one-point advantage and quickly added yet another penalty after Sione Kalamafoni was penalised for attempting to hurdle a pair of London Welsh defenders.
But Burns replied with his third penalty and then superbly created Knight's winning try to give Stuart Lancaster a glimpse of what he could offer come the autumn internationals.
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