Adebayor's 30th-minute header from Gareth Barry's corner was enough to lead Mark Hughes' side to a fourth successive league and cup victory.
But Portsmouth should have claimed their first point of the campaign when substitute David Nugent fluffed a glorious late chance.
Portsmouth handed debuts to new signings Tommy Smith and Michael Brown, while Joleon Lescott made his first Premier League appearance for City since a £24million switch from Everton.
New Pompey owner Sulaiman Al Fahim donned a personalised replica shirt as he took in his first game since completing his protracted takeover, but it was City who made most of the early running.
Pompey were without David James because of a knee injury and stand-in Asmir Begovic threatened to gift City the lead on the quarter hour.
Begovic spilt Barry's teasing free-kick but, with the Bosnian left on his backside, Micah Richards somehow failed to force the ball over the line from four yards before Richard Hughes got back to clear.
City benefited from more slapstick defending to take the lead 15 minutes later as Adebayor was left unmarked inside the six-yard box to nod in Barry's corner.
The game should have been all but over five minutes before half-time when Richards slid in to convert Craig Bellamy's cross, but referee Howard Webb ruled the goal out because Barry was offside, even though he was not interfering with play.
It took until 12 minutes from time for Portsmouth to seriously threaten Shay Given's goal when substitute John Utaka cut in from the right and fizzed a left-foot curler inches wide.
Fellow substitute Nugent should have earned Pompey a point they would scarcely have deserved seconds after coming on.
Brown's lofted ball put the City defence on the back-foot presenting Nugent with a clear strike at goal six yards out, but he hit his shot into the ground and the ball struck Given in the chest and bounced away to safety.