Uganda followed up their surprise 2-0 win over Democratic Republic of Congo in their first game by taking a 12th-minute lead through Emmanuel Okwi before Khama Billiat levelled for Zimbabwe five minutes before the break.
Musona should have equalised before that but fired over from close range and was guilty of an even worse effort in the second half when he hit the crossbar with an open goal from six metres.
Uganda have four points from two games in Group A with Zimbabwe, beaten 1-0 by Egypt in their first game, on one although they are still in the running for a last-16 place.
Egypt were playing Democratic Republic of Congo later on Wednesday.
"It was a difficult game for us, we scored first and after our goal, we didn't withstand the pressure of this game," said Uganda coach Sebastien Desabre.
"Zimbabwe's goal was logical.... in the second half we changed our organisation and it was better and I am proud of our players."
Okwi got the Cranes off to a flying start by turning in the rebound after George Chigova parried Abdu Lumala's shot.
Zimbabwe reacted well, looked dangerous going forward and levelled after Ovidy Karuru's low cross was turned in by Billiat.
The Warriors should have gone ahead five minutes into the second half as Karuru left Musona with an empty goal to aim at but he struck the bar.
Zimbabwe again looked certain to score when Evans Rusike's shot was deflected onto the foot of the post by Denis Onyango but it somehow spun back into the goalkeeper's arms without crossing the line.
In a game of poor finishing by both sides, Uganda also squandered a good chance as Patrick Kaddu fired over from close range.
Musona nearly made an amends for his misses in stoppage time but his shot from the edge of the area flew centimetres wide.
"Today we created many chances and failed to put them away," said Zimbabwe coach Sunday Chibzambwa. "This happens in football."
(Writing by Brian Homewood; Editing by Toby Davis)
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