Vinatieri made kicks of 28, 47 and 23 yards and Hayden's 31-yard score in the first quarter backed Peyton Manning's 185-yard passing performance for the Colts (6-3), who won for the fourth time in five games.
Manning went 20-of-36 through the air without a touchdown or an interception and tight end Jacob Tamme hauled down seven passes for 73 yards in the victory.
Indianapolis forced four turnovers, three of which were converted into 17 Colts points.
Carson Palmer went 31-of-42 passing for 287 yards and two scores but yielded three pick-offs for the Bengals (2-7), who lost their sixth straight.
2014: The Year in Fashion [PHOTOS]
LeBron James reveals his new Nike shoes