Alexander Steen scored a pair against his hometown team as giveaways helped the Blues to an early lead.
David Backes also scored for the Blues (5-1-1), who were playing for the second time in consecutive nights after Thursday's 3-2 shootout win over Chicago.
The game also saw the Jets' top 2012 draft pick taken off on a stretcher.
Defenceman Jacob Trouba missed a hit early in the second period and slammed his head into the boards, bending his neck back at an ugly angle. He didn't move on his own after he crumpled to the ice.
Both benches emptied as worried players gathered around.
Trouba managed a wave as fans shouted his name while he was being wheeled off. A Jets spokesman said later Trouba was taken to hospital for X-rays and was alert, communicating and moving all his limbs.
It energized the Blues, but then Jokinen evened things up at 15:30 when he fired a bouncing rebound past St. Louis backup Brian Elliott, who finished with 26 saves in his first start this season.
The play also gave right wing Matt Halischuk his first point as a Jet for the assist.
Less than three minutes later though, Trouba whiffed on a pass and presented the puck to the Winnipeg-born Steen, as he skated into the slot to put the Blues ahead 2-1.
It could have been worse. The Blues also made Winnipeg's goal posts ring like church bells in the first period.
Play stopped for several minutes early in the second and the arena was silent after Trouba's accident.
Neither team scored on the power play in the first two periods although the Jets only had one chance.
St. Louis had three, two thanks to Kane, who spent some time warming the Winnipeg bench after the second.
Early in the third period another Jets defenceman limped to the dressing room after Mark Stuart slid into his own net.
Steen was perfectly positioned beside the Winnipeg net at 3:34 of the third to one-time a rebound off Backes's glancing shot from the boards and make it 3-1.
But Kane brought the Jets within one at 14:14 when his slapshot from outside the faceoff circle glanced off St. Louis defenceman Roman Polak's stick.
Then Enstrom slammed home a pass from Dustin Byfuglien at 18:06 to tie it up.
Notes: Former Winnipeg Jet Thomas Steen (1981-95), father to Alexander and a Winnipeg city councillor, was on hand for the Blues' second visit to the city since the Jets were reborn in 2011.