The on-loan Manchester City player broke the deadlock midway through the second half after collecting a pass from another Tottenham newcomer in Scott Parker.
After an even opening 45 minutes, Tottenham were the better side in the second period and Jermain Defoe doubled their lead.
Spurs fans may take time to warm to Adebayor after his spell at Arsenal but more of these decisive sort of contributions will surely win them over.
A clean sheet would also have delighted Tottenham boss Harry Redknapp after conceding eight goals in the opening two games, and there was an assured display from the recalled Ledley King at centre-back.
Parker set up an early chance for Defoe after dispossessing Karl Henry 30 yards from the Wolves goal but the England striker's shot was deflected wide.
Spurs were initially the more threatening and twice came close to taking the lead within a few seconds.
First Wayne Hennessey did well to deny Adebayor at close range and the follow-up effort from Niko Kranjcar was cleared off the line by Roger Johnson.
Wolves hit back and Brad Friedel had to be alert to beat out a powerful drive from former Spurs player Jamie O'Hara after Richard Stearman had caught Kranjcar in possession.
The Spurs goalkeeper then excelled in turning a low curling shot from Henry around the post at full stretch after a powerful run by the midfielder.
Wolves were now asking questions and Friedel held onto a Michael Kightly header from Kevin Doyle's cross.
Play was flowing from end to end and King's header bounced down and over the bar from a Kranjcar corner.
Adebayor had his first shot on goal after half an hour but he did not extend Hennessey.
Spurs defender Kyle Walker became the first player to be booked for bringing down Kevin Doyle.
Defoe was looking sharp for the visitors and had a shot turned around the post by Hennessey.
Gareth Bale dragged a shot into the side-netting in time added on.
Henry was being caught in possession too often and another lapse let in Adebayor, but Hennessey was alert and able to hack the ball clear from the on-loan player.
O'Hara tried his luck with a volley from just outside the box but it was easily dealt with by Friedel.
Adebayor managed to get in a shot despite the attention of three defenders but it was deflected over the bar.
But he was not to be denied and put Spurs ahead after 67 minutes. Parker supplied the vital pass and Adebayor took the ball around Hennessey before netting from close range.
Wolves try to retaliate and Parker was booked for bringing down Kightly in a promising position.
Kightly and Stephen Hunt were replaced by Adam Hammill and Matthew Jarvis in a double change by Mick McCarthy.
Hammill was immediately yellow-carded for bringing down Parker before Spurs doubled their lead. Defoe played a one-two with Kranjcar before drilling a low shot past Hennessey.
Johnson showed his frustration by being booked shortly afterwards for chopping down Defoe.