Chabangu's quick brace did the damage early on, while Chenene added the third to hand Swallows an early 3-0 advantage.
Modise scored with a fantastic free-kick in the 23rd minute, but Swallows held onto the two-goal advantage until injury time, when Mushekwi grabbed another.
The teams had played to a 3-3 draw in the first leg in Atteridgeville, and Swallows go through on a 5-6 aggregate.
Swallows will now face SuperSport United in the MTN8 Final on September 22, with the venue still to be determined.
Sundowns had the first good chance of the match inside five minutes, when Modise stepped up to take a free-kick from 30 metres. The Downs captain went for the cross, finding Lebohang Mokoena, whose header went straight to Greg Etafia.
Swallows edged into the game, and on 10 minutes Nomvete made a run down the right, finding Chabangu unmarked for the first goal.
Four minutes later Swallows had their second, as Nomvete again provided the cross for Chabangu, whose cool finish handed The Birds a 2-0 lead.
On 20 minutes Lefa Tsutsulupa made an enterprising run into the box, but his shot was blocked by Wayne Sandilands. Chenene was there to collect the rebound, however, and headed home for the third.
Three minutes later, Modise pulled one back with a free-kick into the top right corner.
10 minutes from half-time Tebogo Langerman almost added a second for Downs, but his free-kick, in a good position on the edge of the penalty area, flew over the crossbar.
Sandilands had to dive full length to deflect a shot from Chabangu 15 minutes from time, as the former Sundowns man searched for his hat-trick.
The Brazilians made a final push in the last 15 minutes of the game, and Mushekwi got their second in injury time, but it proved too little, too late.