The young England prospect booted 19 points, including conversions of both Gloucester tries by wing James Simpson-Daniel and flanker Sione Kalamafoni.
Wasps left the west country with a losing bonus point after centre Eliot Daly's quality try, 11 points for fly-half Stephen Jones and penalties by full-back Tommy Bell and substitute Nicky Robinson.
It was the Londoners' third defeat in four starts this season. Gloucester, though, will head to newly-promoted London Welsh next weekend having lost just once.
And while the home side laboured at times, they ultimately had too much cohesion for a Wasps side that only fleetingly brought their try-machine wings Tom Varndell and Christian Wade into the contest.
Simpson-Daniel provided the game's highlight through a jinking solo score that showcased the 30-year-old's renowned finishing skills at their finest.
And it helped his team rediscover a winning formula on home soil following successive league defeats against Exeter, Newcastle and Sale Sharks last term, and Northampton in this season's opener.
Gloucester boss Nigel Davies handed first starts to former Cornish Pirates full-back Rob Cook and Kalamafoni, while Jonny May featured on the wing as injuries sidelined Charlie Sharples and Shane Monahan.
Wasps were without hand injury victim Hugo Southwell, so Bell deputised at full-back, and England forward James Haskell featured in the back-row instead of Joe Launchbury, who suffered a knock during training earlier this week.
Gloucester started the game at a rapid tempo, sparked by scrum-half Dan Robson's break that was taken on by his forwards and ended with Burns diving over Wasps' line.
But the video referee ruled in Wasps' favour, which helped to settle them down and resulted in a sharp riposte which ended with Jones slotting an eighth-minute penalty.
Burns quickly drew Gloucester level, though, and then after he landed a long-range penalty the home side extended their advantage in vintage Simpson-Daniel fashion.
The former England wing was freed wide out before he left four Wasps defenders clutching thin air during an angled 30-metre run for the 63rd Premiership try of his career.
Burns converted and matters deteriorated for Wasps just five minutes later when an unmarked Kalamafoni pinched possession at the back of a ruck and stormed over for a try, despite Joe Simpson's spirited defensive work.
Burns again added the extras but, with Gloucester still patting themselves on the back, Wasps struck from Jones' restart through an opportunist Daly touchdown that Jones converted.
It was a frustrating score for Gloucester to concede, and Bell further narrowed the gap deep into first-half injury time when he booted a penalty from five metres inside his own half.
Burns opened the second-half scoring by completing his penalty hat-trick and restoring a 10-point advantage, but Gloucester were as indisciplined as their opponents and two more Jones strikes either side of a fourth Burns penalty made it 26-19.
Burns extended Gloucester's advantage to 13 points with another penalty, before Wasps hovered over the self-destruct button when skipper Marco Wentzel was sin-binned for a technical offence with 17 minutes left.
But Gloucester failed to score when Wentzel was off the pitch and they suffered an injury blow in the closing minutes as hooker Darren Dawidiuk limped off.
Gloucester old boy Robinson arrived off the replacements' bench to kick Wasps within losing bonus-point range, which prompted a nervous closing few minutes for the home team, but committed defence kept Wasps at bay.
It was frantic at times for Gloucester and they were helped by Wasps' lack of attacking composure but there was no way through for the visitors, who lacked try-scoring ideas at crucial times.
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