Gyan struck a perfectly weighted through ball from Kwadwo Asamoah wide of Angola's goalkeeper Carlos Fernandez for the decisive goal in the 16th minute.
But the home team squandered several chances to turn around the score and have only themselves to blame for not making better use of their opportunities.
Tall striker Manucho was a major culprit, missing with a close shot that skied high over the Ghana goal on the stroke of halftime and then directing a 73rd minute header wide of goal.
He also had a header saved by Ghana's goalkeeper Richard Kingson in the 30th minute, finding himself unmarked at the back post after a cross but directing the ball straight at the keeper.
Near the end of the match there were two more chances for the hosts as centre back Kali went forward to join the attack.
He missed a header on the stroke of halftime and had a bouncing effort cleared off the line by Isaac Vorsah deep into stoppage time.
"This is what happens in football. I tried my best," said Manucho immediately after the match in a pitch-side television interview.
Ghana's only first-half chance provided the goal, after a visionary pass from Udinese's highly-rated Asamoah. After the interval they bothered the Angola goalkeeper on only two occasions with half chances.
"We did much better than Ghana, especially in the second half, but we couldn't break them down," said Angola's coach Manuel Jose. "We have to congratulate our side on getting to this stage. We just didn't have the luck today."
Ghana, the last Nations Cup host to lose in the last eight 10 years ago, will next play in Thursday's semi-final against the winner of Monday's quarter-final between Nigeria and Zambia in Lubango.
