A contentious penalty allowed Mohamed Aboutrika to register a consolation goal just before half-time, but Egypt will need a major turnaround in the second leg on November 19 if they are to qualify for the World Cup finals for the first time since 1990.
Gyan, who had his first effort on goal inside 11 seconds, opened the scoring after five minutes, shooting home from a narrow angle.
The lead was doubled in the 22nd minute when Michael Essien went on a mazy run, dribbling into the area before being challenged by Egypt captain Gomaa, who managed only to deflect the ball into his own net.
Essien, having played just once for Chelsea this season, produced a dominant midfield showing over the 90 minutes.
Egypt were awarded their penalty four minutes before half-time when Mohamed Salah was bundled down, though the decision from Moroccan referee Bouchaib Al Ahrach was generous.
Aboutrika converted but Ghana restored their two-goal advantage a minute before the break with Waris’s powerful header from Muntari’s free kick.
Muntari’s cross set up Gyan for the fourth in the 53rd minute before the AC Milan midfielder converted from the spot after Waris was brought down by Egypt’s substitute goalkeeper, Ahmed Al Shennawi.
Atsu’s shot from outside the penalty area capped a one-sided occasion.
Ghana reached the quarter-finals of the previous World Cup, in South Africa. Egypt have missed out on the past five finals despite winning an unmatched four African Nations Cup titles in the same period.
(Reporting by Mark Gleeson, editing by Stephen Wood)