West Ham forward Ayew converted in the 32nd minute after Isaac Isinde was adjudged to have fouled Asamoah Gyan.
It was just reward for 2015's beaten finalists, who made their class tell and threatened to turn Uganda's first outing in the finals since 1978 into a chastening experience.
But Milutin Sredojevic's side made an impressive fist of the second period, with influential midfielder Tony Mawejje coming the closest to finding the equaliser their efforts merited.
Ghana boss Avram Grant opted to leave centre-back Jonathan Mensah on the bench and his defence were given something to think about in the fourth minute when Mawejje had an effort saved by Razak Brimah amid a lively start from Uganda.
But Ghana settled and their superior quality started to shine through – Atletico Madrid midfielder Thomas Partey teeing himself up from Jordan Ayew's cutback to volley over.
Jordan Ayew left brother Andre waiting in vain when he burst towards the byline but overhit his cross, while a more accurate centre from the impressive Christian Atsu found captain Gyan straining to head wide.
Andre Ayew prodded tamely at Uganda goalkeeper Denis Onyango before the half-hour but he made no mistake from 12 yards after Gyan caught Isinde napping and drew contact from the centre-back.
Ghana's progress was checked by Rahman leaving the field on a stretcher having fallen to the turf clutching his left leg.
Isinde had the opportunity to atone for his earlier indiscretion in first-half stoppage time, only to head past the post from Farouk Miya's corner.
Following a listless start to the second period, Uganda almost pulled level spectacularly in the 62nd minute.
Mawejje sent a shot whistling narrowly past the top corner after Ghana failed to clear a set-piece.
Joseph Ochaya was also off-target and Geofrey Massa drilled a free-kick into the wall as Sredojevic's side pressed for an equaliser with growing conviction.
Ghana's earlier fluency had all but deserted them by the final 15 minutes and Standard Liege youngster Miya should have done better when he scuffed straight at Brimah.
Daniel Amartey made vital blocks as the clock ticked down and Ghana managed to hold out - Atsu passing up the opportunity to give them breathing space on the counter-attack - although Grant must know improvements will be required against their other group rivals Mali and Egypt.