The Togo international, who joined City from Arsenal in a £25million deal this summer, took just three minutes to pay off the first instalment of his transfer fee.
Then a second from midfielder Stephen Ireland in the 90th minute finally wrapped up the points for Mark Hughes' men.
However, this was far from a stroll in the park for the big-spending Blues, who were pushed all the way by Sam Allardyce's plucky Rovers.
City boss Hughes included £51million worth of new talent in his starting line-up, with Adebayor, Gareth Barry and Kolo Toure all handed their first competitive starts, whilst fellow new boys Carlos Tevez and Stuart Taylor were on the bench.
Rovers, meanwhile, gave full debuts to Lars Jacobsen and Steven N'Zonzi, who cost the club less than £1million, underlining the huge financial gulf between these two sides.
It took City's millionaires just three minutes to send out a powerful statement of intent as the visitors clinically broke the deadlock from their first meaningful attack.
Shaun Wright-Phillips tricked his way past Stephen Warnock on the right-hand side of the penalty area before squaring the ball for Adebayor, who dispatched an unstoppable right-foot shot past Paul Robinson with great aplomb.
With their pre-match gameplan in tatters, it took Rovers a while to regain their composure but, as the half wore on, they began to launch a spirited fightback.
First Jason Roberts fired wastefully over the bar after City's defence had failed to deal with Morten Gamst Pedersen's long throw.
Then minutes later, City keeper Shay Given produced a world-class save, the Irishman diving brilliantly to his left to keep out a Chris Samba header that looked destined for the bottom corner.
Rovers also had a shout for a penalty turned down when Keith Andrews' shot appeared to strike Richard Dunne on the arm.
Sam Allardyce's charges continued to press for an equaliser as the heavens opened in the second half.
Given had to react sharply again to repel a goalbound header from Roberts.
In contrast, little was seen of City as an attacking force, apart from a low drive from Robinho that brought the best out of Robinson.
That proved to be Robinho's last meaningful contribution as Hughes replaced the Brazilian with Tevez in the 69th minute.
City had chances to kill the game off but Ireland and then Tevez failed to take full advantage.
Ireland finally wrapped up the win when he kept his cool to beat Robinson from close range at the death.