The win moves Zimbabwe level with Lesotho on four points at the top of the pool after the latter defeated Namibia 2-1 in the earlier game.
Zimbabwe play their final pool match against Namibia, who have three points, on Tuesday, while Lesotho tackle Angola, who are now eliminated.
Nasama came off the bench to seal the win for Zimbabwe in the second minute of injury-time when it looked as though the tie was headed for a draw.
Jelson had the best chance of the game for the Angolans but blasted his shot over from close-range, while Zimbabwe’s Kudakwashe Mangami had earlier been denied brilliantly by Eduardo.
Earlier, 10-man Lesotho came from a goal down to secure a 2-1 victory over Namibia thanks to a brace from Raboama Koloti.
After a goalless first half, in which Lesotho had Relebohile Mobane sent-off for a two-footed challenge, Namibia took the lead when Romario Hawiseb volleyed the ball home after a counter-attack to give his side the lead.
Koloti had already missed a number of fine chances for Lesotho before he grabbed the equaliser on 70 minutes as he provided a neat finish having been set free behind the Namibia defence.
And six minutes later he had his side in front, turning his defender to drive the ball low past the Namibia goalkeeper.
He might have claimed his hat-trick too, but fire wide of the post after showing good strength to shake off the defender.
Monday will feature four games as Groups A and B are played to fruition with the final matches in the pools kicking off concurrently.
Holders and hosts Zambia are already out of semifinal contention, but will want to defeat Swaziland (kick-off 12h00 local; 10h00 GMT) at the Nkana Stadium to bow out of the competition in style.
That would take them to four points, but short of the top side in the pool, which will be either Uganda or Malawi, who play in a winner-takes-all encounter at the Arthur Davies Stadium at the same time.
Uganda need just a draw to advance as they will have a better goal-difference over the Malawians.
The fact that Egypt and South Africa both already have six points in Group B means that Zambia cannot finish as the best-placed runner-up either.
The former duo also meet in Group B (kick-off 15h00 local; 13h00 GMT) at the Nkana Stadium where the victor will seal their place in the semifinals.
As they have identical records in terms of points, goal-difference and goals-scored, a draw means it would come down to Fair Play considerations where the team with the fewest red/yellow cards would advance. If that is the same too, then there will be a drawing of lots to see who finishes top of the pool.
The other game in the pool between Mauritius and Mozambique that kicks-off at the same time is a dead rubber, with neither side able to pick up a point as yet.
Namibia 1 (Hawiseb 64’) Lesotho 2 (Koloti 70’, 76’)
Zimbabwe 1 (Nasama 90’) Angola 0
12h00 (10h00 GMT): Zambia vs Swaziland – Nkana Stadium
12h00 (10h00 GMT): Malawi vs Uganda – Arthur Davies Stadium
15h00 (13h00 GMT): South Africa vs Egypt – Nkana Stadium
15h00 (13h00 GMT): Mauritius vs Mozambique – Arthur Davies Stadium