Broxham scored the winning goal in the 83rd minute, collecting the ball after Muangthong had cleared a Victory corner, before rifling his shot past the visiting goalkeeper with the help of a deflection.
After surviving the opening 20 minutes - more by luck than design - Muangthong had taken the lead when Mario Durovski's corner went straight into the net, evading both Victory full-back Scott Galloway at the near post and goalkeeper Nathan Coe.
Melbourne's best chance to equalise in the first half came on the stroke of half-time but captain Mark Milligan failed to convert a penalty.
However, second-half goals from James Troisi and Broxham saw Victory recover and earn a berth in the ACL group stage for the fourth time in the club's history.
Melbourne started brightly and should have taken the lead in the eighth minute only for Tom Rogic to head wide.
Muangthong then gave the hosts an idea of the threat they would pose on the counter-attack throughout the match, when Teerasil Dangda got in behind Victory's defence but he scuffed his shot wide.
Victory continued to dominate but struggled to seriously test the Thai club's goalkeeper Visanusak Kaewruang regularly enough.
And Kevin Muscat's men were punished for their lack of cutting edge when Durovski curled his corner over Galloway's head and inside the near post in the 22nd minute with Coe stranded behind an opposition player.
The half ended in farcical circumstances as Victory were awarded a penalty after Archie Thompson was knocked over in the box by Piyaphon Buntao.
Referee Masaaki Toma from Japan showed Piyaphon both a yellow and a red card, mistakenly believing the defender had already been booked, but quickly rescinded the sending off when reminded of his era by several Muangthong players
Milligan then stabbed his spot kick low to Visanusak's left hand but it lacked power and the visiting goalkeeper held onto the ball.
The drama did not end there either with the referee's whistle for half-time being temporarily mistaken by Victory as the signal for a re-take.
Muangthong should have doubled their lead just after the break but Teerasil failed to convert a one-on-one opportunity, while Durovski's fizzing volley soon after was tipped wide by Coe.
Victory took advantage of those two misses in the 58th minute when Troisi timed his run perfectly to get onto Galloway's cross and headed the ball into the roof of the net.
Melbourne continued to push throughout the second half but missed a hatful of chances and the match seemed destined for extra time, until Broxham's late heroics saw Victory progress.
To add insult to injury, Muangthong ended the game with 10 players after their captain Datsakorn Thonglao was sent off for head-butting Broxham in the final minute.
Melbourne Victory 2 (Troisi 58 Broxham 83)
Muangthong United 1 (Durovski 22)
Crowd: 8304 @ Simonds Stadium
