Kallisto Pasuwa will be elated by his side's performance as they now top group L - unbeaten with eight points after four games played.
The first half was dominated for large spells by the home side as Swaziland looked to contain Pasuwa's attacking line-up and hit them on the counterattack, but Zimbabwe always looked more likely to score - despite Knowledge Musona and Nyasha Mushweki wasting a few guilt-edged chances as the tie went in goalless at the break.
At half-time Sundowns forward Khama Billiat was brought on for Marshall Mudehwe and the in-form attacker made all the difference.
Billiat mad an immediate impact as he sent a free-kick straight to the feet of Musona - the former Chiefs striker struggled to get the ball away and hit it straight at a Swazi defender but shortly after they were awarded a penalty in the 52nd minute for their sustained pressure on Sihlangu Semnikati.
Kudakwashe Mahachi played a quick one-two that led into the Swazi penalty area, the Golden Arrows winger was brought down with the home side erupting to the whistle of referee Ali Adelaïd from the Comoros Islands.
Knowledge Musona converted the spot-kick with relative ease as he cheekily chipped the ball into the back of the net to open the scoring for his side.
It was one-way traffic after the opening goal - Billiat was involved with the second as he sent a dangerous free-kick into the penalty area in the 60th minute for Sparta Prague defender Costa Nhamoinesu to thump in a powerful header to double their advantage.
The crowd was in full voice as Billiat and Mahachi turned on the flair to leave the Swazi midfield chasing shadows. Nhlanhla Gwebu made good saves from Mushekwi and Musona before the former was replaced by Maritzburg United striker Evans Rusike in the 70th minute.
The move looked to be another inspired substation for Pasuwa as two minutes later the in-form striker was involved in their third goal.
Mahachi went on a trademark run through the heart of the Swaziland midfield and defence and played an incisive pass to Rusike, the forward's first touch let him down but the ball deflected off him and past the Swazi goalkeeper in an attempted clearance from a Swazi defender.
Billiat, who was scintillating from his introduction, was rewarded for his efforts as he finished off the scoring in the 87th minute - the Sundowns forward received the ball fortuitously just inside the area and nonchalantly stroked it into the bottom right corner of Gwebu's goal to spark wild celebrations all round the Rufaro Stadium.