Swansea advanced to face holders Portsmouth in the FA Cup fourth round after surviving a late rally at Histon.
Gorka Pintado opened the scoring when the non-leaguers gave him excessive space, the Spaniard having time to turn and tee-up a 25-yard right-foot blast.
Andrea Orlandi tormented debutant Marlon Patterson on the flank, and his cross let Guillem Bauza stroke home.
Pintado missed a late penalty, and soon after Josh Simpson struck to give the Swans a nervous last few minutes.
606: DEBATE
There were plenty of worried Swansea supporters, their club having gone out to non-league Havant and Waterlooville in last year's third round.
But Roberto Martinez's men held out to extend their unbeaten run to 10 games and to set up a fourth-round trip to Fratton Park.
The build-up to the match at the Glass World Stadium had been controversial as the original date had been postponed due to a frozen pitch.
Blue Square Premier side Histon - in the third round for the first time - accused Swansea of putting pressure on the Football Association to make the decision, although they later apologised.
Histon manager Steve Fallon:
"We had three or four good chances in the first half and might have taken them.
"When they come along in games like these you've got to take them. Swansea had a lot of quality about them, and their finishing was very good.
"But it's a credit to the lads that they are disappointed to have lost to a Championship team.
"Now comes the hard part of picking the players up when all the publicity ends."
Swansea manager Roberto Martinez:
"I'm extremely proud, the players had to adapt to the conditions, it takes a lot of character and maturity.
"Andrea Orlandi has had a stop-start season with injuries, it's great to see him out there enjoying himself - he's one of those players who puts a smile on your face.
"I'm looking forward to Portsmouth, more than anything for the fans. We haven't had that glamour for a few seasons.
"It's great to play the cup holders in any competition. It's a great challenge and a glamorous tie to look forward to."
Histon: Naisbitt, Oyebanjo, Langston, Ada, Patterson (Bygrave 46), Murray, Mitchell-King, Simpson, Knight-Percival, Wright, Reeves (Andrews 46).
Subs Not Used: Welch, Pope, Kennedy, Barker.
Booked: Ada, Naisbitt.
Goals: Simpson 84.
Swansea: De Vries, Serran, Monk, Williams, Tate, Orlandi (MacDonald 76), Tudur-Jones (O'Leary 56), Britton, Butler (Dyer 64), Pintado, Bauza.
Subs Not Used: Pennock, Scotland, Gomez, Rangel.
Goals: Pintado 22, Bauza 38.
Att: 2,821
Ref: Phil Crossley (Kent).
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