Jadeja's disciplined left-arm spin led Chennai's bowling charge to restrict Kolkata to 137-9, a total the home team overhauled with 14 balls to spare at M.A. Chidambaram Stadium.
Two-time winners Kolkata suffered defeat after three straight victories.
Skipper Ruturaj Gaikwad led the chase with an unbeaten 67 and a 70-run second-wicket partnership with Daryl Mitchell, who made 25, to make Chennai bounce back from two losses.
"Jaddu (Jadeja) always comes in after the power play with momentum in the spin department," Gaikwad said of his man-of-the-match performer.
"With this team, I don't really need to tell things to anyone. Everyone's in a great head space, Mahi (M.S. Dhoni) bhai and (Stephen) Fleming are still around to take those calls," Gaikwad added.
Gaikwad, who was handed the captaincy by veteran Dhoni ahead of the season, hammered nine boundaries including the winning hit to make it three in three at home this season.
Shivam Dube hit 28 off 18 balls before departing with Chennai needing three for victory and the crowd erupted at the sight of Dhoni, who led the team to five IPL titles, walking out to bat.
The 42-year-old, who is likely playing his last season as player, made one off three balls amid loud cheers from the fans dressed in the Chennai's yellow.
Kolkata had a horror start to their innings after they lost England's Phil Salt on the first ball off fast bowler Tushar Deshpande.
Jadeja remained the hero as he struck on the first ball of his opening over and sent back the in-form Sunil Narine (27) four balls later to put Kolkata in trouble at 60-3 inside seven overs.
Wickets kept tumbling as Jadeja, who reached 100 catches in the IPL, took one more and fellow spinner Maheesh Theekshana reduced the opposition to 85-5 in 11.5 overs.
The innings never took off with skipper Shreyas Iyer adding some respect with his 34 in 32 balls.
"We have to go back to our drawing board and learn, just a matter of one match and one innings, glad it happened at the start of the tournament," said Iyer.
Deshpande claimed three wickets and Bangladesh left-arm quick Mustafizur Rahman took two for Chennai.
