Winnipeg goaltender Ondrej Pavelec faced four consecutive must-save shots to keep the Jets' hopes alive in the shootout before Jokinen put one past Brian Elliott. Pavelec then stopped Chris Stewart to ensure the victory.
St. Louis lost for just the second time this season and suffered its first overtime setback.
Alexander Steen scored twice for the Blues, the second of those goals giving them a 3-1 lead 3:34 into the final period. Steen has seven goals for the year, tying him for most in the NHL with San Jose's Tomas Hertl.
Jokinen scored late in the first period to bring about a 1-1 tie.
Pavelec made 29 saves in regulation and overtime before stopping 5-of-7 in the shootout. Stewart stopped 24 shots.