The Dutchman got down low to his right to make the stop after captain Ashley Williams gave away a 73rd-minute spot-kick for a foul on Jordi Gomez.
It earned the hosts their first point in the Premier League and they thoroughly deserved it after dominating large parts of Saturday's game, with Danny Graham having a series of chances and Scott Sinclair firing wide at the Liberty Stadium.
But they also rode their luck as second-half efforts from Gomez and Victor Moses came back off the crossbar with Vorm beaten.
Alan Tate dropped to the bench for the hosts with left-back Neil Taylor available again following suspension, while Wayne Routledge made his full debut in place of Stephen Dobbie.
Antolin Alcaraz was the only change for Wigan as manager Roberto Martinez returned to his former club, the Paraguayan coming in for Adrian Lopez.
Swansea started brightly and some good link play from Sinclair and Angel Rangel put the latter away on the right. His cross was half blocked with Graham's shot being sharply saved by Ali Al Habsi.
The hosts' lively opening continued as Graham dragged his effort after racing through on goal, with Sinclair recovering the wayward shot to cut inside and fire wide of the near post.
Graham was soon involved again as the record signing headed wide from Kemy Agustien's cross.
But it took a great tackle by Williams to deny a surging Franco Di Santo run as the Wigan striker closed in on goal.
The hosts continued to dominate possession and James McCarthy covered back well to block off a Taylor pull back.
Alcaraz then brilliantly denied Graham a tap-in after Sinclair and Routledge combined down the left, but the defender soon departed the fray as a calf problem picked up early on saw him withdrawn, with Ronnie Stam coming on.
Moses fired a tame effort wide before Gomez worked Vorm in the Swansea goal for the first time with a shot from an acute angle just before the break.
Mohamed Diame was off target from distance shortly after the restart but Swansea were soon back on the attack, Sinclair blazing high and wide after Graham had robbed Stam.
Rangel's scuffed effort was directed into Graham's path just six yards out in the 54th minute but the former Watford man was crowded out by Latics defenders.
But Wigan were posing a greater threat than in the first 45 minutes and the dangerous Moses had an effort diverted wide by an Agustien block, before Routledge was narrowly off target with a shot across Al Habsi at the other end.
Moses tested Vorm after the Swans defence failed to pick him up at a short corner, before Gomez struck the woodwork with a lovely outside of the foot effort from the edge of the box as the visitors enjoyed their best period of the game.
Swansea's failure to capitalise on their dominance looked as though it would cost them when Wigan were awarded a 73rd minute penalty after Williams brought down Gomez as the midfielder followed up a Moses shot that had also cannoned back off the bar.
But Vorm made himself an instant hero with the Liberty Stadium faithful as he got down low to his right to keep out Watson's spot-kick.
Substitute Hugo Rodallega then somehow failed to connect with a header back across goal from the resulting corner.
Swansea replacement Dobbie's shot from the edge of the area was then parried away by Al Habsi, but Sinclair could not properly connect with the rebound as the game entered the final 10 minutes.
Home nerves became frayed as the Latics finished strongly with Maynor Figueroa heading across the face of goal, but Swansea held out for a point.