The Swans fell to a third consecutive defeat as Spurs claimed the league double over Michael Laudrup's side at the Liberty.
Jan Vertonghen showed superb quality to fire the visitors ahead after slotting past Vorm following a sublime pass from Gareth Bale, before the Welshman then blasted into the top corner to leave the Swans stunned.
Nathan Dyer hit the crossbar with a header before the break, and the Swans showed more threat in the second half.
Michu headed home sub Ki Sung-Yueng's corner with 19 minutes to go, and that set up a tense finish.
Bale and sub Jermain Defore went close for Spurs while Michu and then Dyer probed for an equaliser, but that never came as Spurs held on to move up to third.
The Swans made three changes to the starting line-up. Angel Rangel missed out with a calf strain, which meant Dwight Tiendalli came in while Chico Flores made his return from the ankle ligament injury which had kept him out since the home win over QPR on February 9.
Wayne Routledge was also restored to the starting XI as Ki Sung-Yueng was included among the replacements.
The visitors were without injured keeper Hugo Lloris, so American Brad Friedel started between the sticks.
Former Swans loanee Gylfi Sigurdsson was included in Andre Villas-Boas' midfield while Steven Caulker was among the substitutes.
Swans: Michel Vorm, Dwight Tiendalli, Chico Flores (Garry Monk, 67), Ashley Williams (capt), Ben Davies, Leon Britton, Jonathan de Guzman (Ki Sung-Yueng, 57), Wayne Routledge, Nathan Dyer, Pablo Hernandez (Itay Shechter, 80), Michu.
Subs: Gerhard Tremmel, Bartley, Roland Lamah, Luke Moore.
Spurs: Brad Friedel, Jan Vertonghen, Aaron Lennon, Scott Parker, Emmanuel Adebayor (Jermain Defoe, 61), Gareth Bale, Kyle Naughton, Mousa Dembele, Michael Dawson (capt), Gylfi Sigurdsson (Lewis Holtby, 76), Kyle Walker.
Subs: Jordan Archer, Tom Huddlestone, Jake Livermore, Benoit Assou-Ekotto, Steven Caulker.
The Jack Army were looking for a positive start from the Swans after two consecutive defeats, but it was the visitors who made the dream start.
After just eight minutes, centre-back Jan Vertonghen raced forward before collecting a return pass from Gareth Bale. He brought down the chipped pass excellently before slotting past Michel Vorm from ten yards.
Things could have got worse for the Swans just five minutes later, Emmanuel Adebayor racing in on goal after a through ball, but he scuffed his effort straight at Vorm.
The Swans replied via Michu as the Spaniard rose highest to meet Jonathan de Guzman's corner, but he couldn't keep it on target.
But Spurs delivered another massive setback in the 21st minute, Vertonghen fooling everyone with a cutback pass to Bale, who took one touch before rifling his effort past Vorm into the top corner.
The Swans needed an instant reply with Nathan Dyer latching onto a through ball, but his attempted chip floated wide.
Dwight Tiendalli then hammered an attempt at Brad Friedel, with the American stopper swatting it wide for a corner, before Ben Davies fired a right-footed effort high and wide.
The chances kept flowing, Bale waltzing past Chico Flores and Leon Britton before curling an effort on the outside of his left foot just past the post.
Then came the Swans' best chance of the game after 35 minutes, Tiendalli's perfect cross finding Dyer, but his header cannoned off the crossbar and deflected back off him and out for a goalkick.
Half-time: Swans 0 Spurs 2
The visitors made the brighter start to the second half, and it took a fine reflex save from the outstretched Vorm to deny Gylfi Sigurdsson, before the Dutchman quickly got up to deny Bale from the rebound.
The Swans responded again via Michu, who rose to meet Tiendalli's fine cross, but the striker's header bounced a yard wide.
Michael Laudrup made his first change before the hour, Ki Sung-Yueng replacing de Guzman, with Garry Monk coming on in place of Chico shortly after.
The change paid off, with Ki's outswinging corner finding Michu, who headed in via the back post after 71 minutes.
It raised the atmosphere up a notch at the Liberty, but the cheers had barely settled down by the time Bale ran on 50 yards before firing into the sidenetting on Vorm's near post.
Michu then found the sidenetting after Ashley Williams flicked on Tiendalli's cross before Laudrup made his final change as Itay Shechter came on for Hernandez in the last ten minutes.
Sub Jermain Defoe then brought the best out of Vorm, driving low across goal which drew a great save from the keeper as the game entered the last five minutes.
Then the drama raised another level as Routledge's cross dropped to Dyer, but his volley was blocked superbly by former Southampton team-mate Bale inside the five minutes of allocated time.