The Welsh side squandered a glorious chance to break the deadlock with six minutes remaining when they were awarded a spot-kick after Canaries goalkeeper John Ruddy brought down Scott Sinclair.
But the summer signing from Everton redeemed himself by making a stunning save to keep out David Cotterill’s effort.
Three minutes later Norwich – who beat Scunthorpe 1-0 last Saturday courtesy of Grant Holt’s 90th-minute header – left it late again the claim the spoils.
Second half substitute Anthony McNamee sent over a cross which Williams diverted into his own net.
The Canaries wrapped up the points deep into injury time when Holt teed up substitute Jackson, who opened his account following his summer move from Gillingham from 12 yards.
Swansea, who have gone six games without an away win and three without a goal, had two efforts at goal inside the opening 15 minutes but Sinclair and Nathan Dyer failed to find the target from distance.
Norwich, who claimed their first win on their return to the Championship at Glanford Park a week ago, responded and midfielder Korey Smith forced Swansea goalkeeper Dorus De Vries into a save from 18 yards.
After Cotterill had sent a 35-yard lob wide, the home side came close again in the 25th minute.
A corner from Andrew Surman – making his 200th career appearance – was met by Andrew Crofts, whose close-range header went just wide.
The Swans, though, finished the first half on top with Darren Pratley headed over Cotterill’s corner with the latter also drilling a shot wide from 20 yards.
Either side of that opportunity, Stephen Dobbie – who helped Blackpool secure promotion to the Barclays Premier League last season – twice came close to scoring for the second successive game but could also not trouble Canaries goalkeeper Ruddy.
Norwich playmaker Wes Hoolahan had a 25-yard shot saved from De Vries as the home side made a bright start to the second half before Swansea squandered a great chance to take the lead in the 54th minute.
Sinclair pounced on a loose ball and raced clear with only Ruddy to beat but the former Everton stopper pulled off a stunning save to deny the winger.
Swansea kept pressing and Dobbie smashed a shot wide before seeing another effort saved by Ruddy while Hoolahan put a shot wide from 18 yards at the other end.
But Cotterill missed a great chance when his spot-kick was saved by Ruddy and Norwich punished them with Williams’ own goal being followed by Jackson’s effort at the death.