The Colts (9-4) were trailing 23-21 with 6 1/2 minutes left in the contest when Vinatieri nailed his go-ahead 53-yarder and then added some insurance with another field goal less than 3 minutes later.
Andrew Luck managed to complete just 16-of-34 passes for 196 yards with a touchdown and two interceptions, but still overcame a 13-point halftime deficit as Indianapolis won for the seventh time in eight games.
Vick Ballard contributed 94 rushing yards on 19 carries as the Colts improved their playoff chances.
Reggie Wayne made six grabs for 64 yards with a touchdown for the victors.
Tennessee (4-9), meanwhile, dropped its third straight despite a 262-yard, 22-of-35 performance from Jake Locker, who also tossed two second-half interceptions.
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'