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Vinatieri edged the visitors into the lead with his second kick of the game, while the Colts scored the first touchdown of the match just before the break when Blake Bortles fumbled the ball from a snap into the endzone and Robert Mathis gathered for the score.
The home side came out the blocks the fastest in the second half and Bortles threw his first touchdown pass of the game when he looped the ball down the sideline to Allen Hurns to race past his man for an 80-yard touchdown to put Gus Bradley's men ahead.
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Vinatieri continued his fine day for the Colts by firing his third kick of the day through the uprights, but the Jaguars took command of the contest when Bortles found Allen Robinson in the endzone following an impressive rout from the wideout.
Bortles found his rhythm at the start of the fourth quarter and secured the victory for his team by firing to Julius Thomas on a fade route for a three-yard score.
Denard Robinson poured on the agony for the visitors when he raced through the line to take his team to the goalline before he finished off the move himself by finding a gap to complete the score.
The Jaguars reached the 50-point mark for the first time in their history when Bortles jumped over the goalline himself for a one-yard touchdown to cap an impressive second-half display from Bradley's team.
The result knocks the Colts out of first place in the AFC South, while the Jaguars improve to 5-8 in their season.