The 23-year-old former Scotland Under-19 captain - signed from Edinburgh - sparked a fine recovery by the Sharks, who dominated for long spells to turn the match around after trailing 10-3 after 14 minutes.
Sale's other try came from skipper David Seymour with 19 points from the boot of fly-half Nick Macleod, relishing the role of front-line kicker following Charlie Hodgson's move to Saracens.
Macleod took his points tally for the campaign to 41 from three matches with five penalties and two conversions as Sale notched their second victory of the season.
Former Sale centre Chris Bell and fly-half Ryan Davis scored the Wasps tries, with Davis adding a conversion and two penalties.
Macleod kicked Sale into a third-minute lead after Wasps were caught offside but the hosts responded five minutes later with an excellent try.
Full-back Richard Haughton, making his first start of the season, provided the springboard for a four-man move, fielding a kick just inside his own half and bursting through a gap in the Sale midfield.
He found England Under-20 flanker Joe Launchbury up in support before Wasps' two ex-Sale centres provided the finish, Chris Mayor producing the final pass for Bell's touchdown.
Davis converted and stretched the lead to 10-3 with a 14th-minute penalty which went in off the far post.
Macleod was less fortunate two minutes later when his penalty rebounded from the post.
But he quickly made amends with a much more difficult kick, striking his third penalty shot superbly from the halfway line in the 20th minute before hitting the target again 10 minutes later to cut the deficit to one point following a Wasps' scrum infringement.
Sale, orchestrated by Wales' World Cup scrum-half reject Dwayne Peel, stepped up the pace towards the interval.
Full-back Rob Miller and winger Joaquin Tuculet combined down the left to get within a metre of the Wasps line and set up a spell of pressure which led to their side's opening try.
Hooker Marc Jones and flanker David Seymour were both held up just short before McKenzie forced his way over for his first try for the Sharks following his close-season move from Edinburgh.
Macleod landed the conversion and maintained his superb kicking form with his fourth penalty following another Wasps scrum offence on the stroke of half-time.
Sale should have increased their lead two minutes into the second half when centre Will Addison left Wasps hooker Rob Webber standing, cut inside him and found Charlie Amesbury outside him but the winger knocked on under pressure as he dived for the line.
Amesbury was denied again when he was flattened by a huge try-saving flying tackle from England prop Tim Payne before James Gaskell was stopped tantalisingly close to the line by weight of numbers.
Sale lost the influential Peel, replaced by Scott Mathie in the 51st minute, but the change did not affect their momentum.
After the Wasps scrum was penalised yet again, the Sharks spurned the chance of a penalty kick to kick for touch and were rewarded with Seymour's try.
Kearnan Myall took the lineout catch and his forward colleagues produced an excellent rolling maul to drive over the line with the number seven grounding the ball and Macleod converting.
Wasps responded with a 57th-minute penalty from Davis, cancelled out five minutes later by Macleod.
The home side managed a late attacking flurry with Davis grabbing a fine solo try, jinking and turning to squirm his way over in the 68th minute, only to miss the conversion.
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