
Asamoah Gyan scored the other goal for the hosts, who had three shots hit the upright and a goal disallowed, while Oumar Kalabane scored for the outsiders.
The late winner sent the 40,000 capacity crowd in the Ohene Djan stadium and tens of thousands more gathered outside the venue into celebrations, and the streets filled the minute the referee blew the final whistle.
Drivers throughout the capital blew their horns as passengers hung out of windows waving flags.
Ghana coach Claude Le Roy said that it was never easy to win the first game, but he was pleased that they had managed to pick up their first three points in the tournament.
However, he criticised the condition of the pitch.
"I have been coaching in Africa for 20 years and never before have my teams played on a field like this," Le Roy said. "It made it very difficult for my side, who rely on one-touch football."
Both teams started their Group A match briskly, but it was the home side that had the first chance in the fifth minute, when Muntari sent through Gyan, but the striker placed his shot wide of goalkeeper Kemoko Camara's goal.
Junior Agogo came even closer to opening the score in the 19th minute, when his powerful header from a cross by Quincey Owuso-Abeyie left Camara flat-footed.
The Nottingham Forest player's header crashed against the right upright and trickled along the goalline before being cleared by a defender.
Guinea were saved by the post again in the 27th minute when Owuso- Abeyie, who was only cleared to play for the Black Stars by the sport's controlling body FIFA shortly before the tournament started, having earlier represented the Dutch youth team, also headed against the post.
The Black Stars had the ball in the back of the net in the 36th minute, but referee Eddy Maillet from the Seychelles ruled that Gyan had pushed a defender.
A few minutes later, it was Muntari who came close, but his shot from close range again hit the upright.
Much to the delight of the home crowd, who were waving flags and beating drums throughout the game, Maillet awarded the home side a soft penalty in the 54th minute, when he rather harshly adjudged Kalabane to have brought down Agogo in the area. Gyan made no mistake from the spot.
Guinea captain Bobo Balde, who plays for Scottish club Celtic, said that he thought the penalty decision was not fair.
"The Ghana player went down without being fouled. But then we got back into the game," Balde said.
"We have nothing to be ashamed about, and we can still qualify if we do well in our next two matches against Morocco and Namibia. There is no reason why we should not do well."
Guinea's Kalabane made amends in the 65th minute when he headed in from Pascale Feindounou's corner.
Substitute Andre Ayew nearly won the game for the Black Stars in the 77th minute with a well-taken shot from close range, but Camara pulled off a sensational save to deny the son of former African Footballer of the Year Abedi Pele.
"It was a fantastic feeling coming on for my first competitive match for my country. I am so pleased that we started with a win, and now we can go on to achieve a lot in this tournament," Ayew said.
The Guineans were denied a share of the spoils in the last minute when Muntari's shot from outside the box gave Camara no chance.
The other two teams in Group A, Morocco and Namibia, will face each other Monday in the same stadium. Also Monday, Nigeria meet Ivory Coast and Mali face Benin in two Group B matches in Sekondi, Ghana.
By Peter Auf der Heyde, dpa