Collins’ powerful header with just over 15 minutes to play was the decisive goal in a game in where tempers flared as Swansea’s Jordan Garrick was shown a red card for throwing a punch towards the goalscorer.
The Hatters were good value for all three points as they stretched their unbeaten run to five matches and moved to within two points of safety.
Jones made three changes to the side that earned a point against Preston North End last weekend, with captain Sonny Bradley returning as well as Jacob Butterfield and Elliot Lee.
The latter was heavily involved in Town’s bright start to the game with some neat touches and interlinking play.
Simon Sluga stood tall to block Andre Ayew’s goal-bound effort in the 14th minute and the Hatters then made a break down field as Harry Cornick skipped past the challenge of Jake Bidwell down the right, made his way into the box, cut back on to his left foot but blazed over the bar.
Town continued to impress in the first half and Collins was inches away from latching onto Dan Potts’ cross five minutes before half time, as Woodman gathered the ball just ahead of the striker in the six-yard box.
Swansea came out the better side in the second half initially as Ayew went close once more but Sluga was quick off his line to smoother the shot.
Just before the hour mark Pelly-Ruddock Mpanzu had the Town’s first shot on target from distance but Woodman was able to gather easily.
The game was far more end-to-end as play continued with Swansea coming close to taking the lead. Matt Grimes swung a corner in from the right which Ben Cabango got his head to, but he was denied brilliantly by Sluga, before Brewster headed over the rebound.
Shortly after the second half drinks break, and just after sub James Bree had almost made an immediate impact with a shot that Woodman saved, the Hatters took the lead in the 73rd minute through Collins’ 12th goal of the season.
Izzy Brown, also on as a game changer, produced an inch-perfect cross and the striker rose highest to power his header into the bottom corner despite Woodman getting a hand to it.
Tempers flared with ten minutes remaining as challenges went flying in and Garrick was sent off for aiming a punch at Collins after a huge melee just in front of the Hatters’ technical area.
There were nine minutes added on at the end of the match and the Hatters withstood late pressure from the hosts to come away from the Liberty Stadium with all three points.
SWANSEA: Woodman, Roberts, Cabango (Routledge 86), Wilmot, Bidwell; Fulton (Byers 58), Grimes; Kalulu (Garrick 45), Gallagher, Ayew, Brewster.
Subs not used: Mulder, Guehi, Rodon, Naughton, Dhanda, Dyer.
Yellows: Wilmot, Gallagher
Red: Garrick
TOWN: Sluga, Pearson, Carter-Vickers, Bradley (C), Potts (Cranie 90+7), Butterfield (Brown 59), Rea, Mpanzu, Lee (Bree 70), Cornick (Hylton 90+7), Collins.
Subs not used: Shea, Tunnicliffe, McManaman, Shinnie, Moncur.
Yellows: Butterfield, Rea, Mpanzu
Goal: Collins (73)
Referee: Andy Woolmer