English referee Mike Dean handed out eight cards, one of them red, in a feisty encounter that was settled by Damiao's cool 45th minute strike.
The Internacional striker, who Spurs were interested in signing this summer, also finished off a brilliant passage of play started by the excellent Neymar but the linesman flagged the 22-year-old offside just after he had tapped the ball past Ghana stopper Adam Kwarasey.
The result eases the pressure on Brazil boss Mario Menezes, who came under fire for his team's quarter-final exit from the Copa America.
His recall of Ronaldinho, making his first appearance in 10 months, also paid off as the 31-year-old offered glimpses of brilliance and came close to scoring with an outstanding free-kick just before full-time.
A wall of noise from the near-capacity crowd greeted the players as they kicked off at the home of Fulham and it did not stop until long after the final whistle.
Ghana applied the early pressure as Jonathan Mensah headed over the second of two early corners for the Africans.
Brazil's Elias replaced Ganso nine minutes in and he went in to Dean's book 30 seconds later after clattering in to Derek Boateng.
The booking marked the start of an ugly period that saw Dean dish out six cards by half time.
Daniel Opare was the second to see yellow, leaving Neymar in agony with an ugly challenge. Boateng and Fernandinho joined him before Opare was sent off just after the half hour when he was harshly adjudged to have clipped Lucio on the follow through to receive a second yellow.
Shortly before, Brazil thought they had taken the lead when Neymar shimmied past two Ghana defenders, scooped the ball over his marker for Damiao to nip the ball past Kwarasey.
The Brazil support broke in to dance, but the linesman's flag cancelled the brilliant effort out.
Ghana's terrible indiscipline continued as Mensah became the fourth Black Stars player to be booked and Brazil started to take command of the game.
The South Americans took the lead in the last minute of the first half when Fernandinho cut the Ghana defence open and Damiao kept his nerve to slot past the on-rushing stopper.
Damiao should have added his second just after the break but he could not get a foot on Hulk's cross.
Former Fulham favourite John Pantsil put in a brilliant block to deny Marcelo before Anthony Annan was booked for a cynical hack on Neymar and Samuel Inkoom followed suit by barging over the Santos hitman and earning himself a booking.
Ronaldinho, who had perked up after a quiet first half, drew a good save from Kwarasey with 10 minutes left.
AC Milan star Pato came off the bench and saw a well-timed header superbly tipped over by Kwarasey.
The Ghana stopper had to be at his best to tip Ronaldinho's excellent free-kick around his left-hand post as Brazil looked for a second with time running out.
The former Barcelona man whipped over a teasing cross that was hooked away by a Ghana defender in injury time and the Brazilians had to settle for the one-goal victory.