Aurelien Collin, who scored Sporting's equaliser in the second half, put his side ahead in the shootout with a thunderous shot inside the post to beat Real goalkeeper Nick Rimando.
Real's Lovel Palmer then hit the crossbar with his kick to send the home crowd into rapturous celebration on a freezing night at Sporting Park in Kansas City.
Collin was named the game's MVP for his role in the victory.
"It's an incredible feeling," Sporting defender Matt Besler said. "We've been working at this for a long, long time.
"I'm proud of the guys, I'm proud of the fans and the city.
"It was a grind, the weather didn't play anybody any favors as the field got frozen there but Jimmy came up big for us and we are the champions."
Salt Lake opened the scoring in the 52nd minute after Kyle Beckerman found Alvaro Saborio with a pinpoint pass and the Costa Rican chested the ball down before beating Kansas City goalkeeper Nielsen with a powerful low shot.
Sporting, who previously hoisted the giant silver cup in 2000, leveled in the 76th minute when Collin headed the ball inside the far post after a corner kick from Graham Zusi.
Saborio thought he had scored with a header in extra-time only for it to be ruled offside, much to the delight of the home crowd.
Real missed their opening two penalty kicks in the shootout to trail 2-0 before Besler missed the home side's third attempt.
Zusi had the first opportunity to win the championship for Sporting but fired over the bar to maintain hope for the visitors.
Real were then given their own chance of victory after Lawrence Olum missed wide left but Nielsen's diving save against Velasquez proved pivotal in setting up Sporting's win.
It marked the third time the MLS Cup was decided in a shootout following RSL's win in 2009 and Houston Dynamo's win over New England Revolution three years earlier.