Skipper Garry Monk marked his comeback with an injury-time winner to send the Swans into the fourth round of the Capital One Cup.
Crawley pushed the Swans all the way and the visitors had to come from behind with goals from Michu and Danny Graham.
But it was Monk who stole the show on his first start of the season following a back injury.
Swans boss Michael Laudrup made seven changes to the side that lost 3-0 at home to Everton on Saturday.
Goalkeeper Gerhard Tremmel, skipper Garry Monk, Luke Moore, Dwight Tiendalli, Mark Gower, Leon Britton and Jazz Richards were all given starts.
From Saturday, Michu, Wayne Routledge, Ben Davies and Ki Sung-Yueng remained, with the South Korean starting at centre-half alongside the returning Monk.
New signing Tiendalli made his debut at right-back with Davies on the left. Britton, Gower and Michu started in midfield, with Richards and Routledge out wide. Moore took the central striker’s role with Danny Graham on the bench along with Alan Tate
and members of the club’s Under-21 squad.
League One Crawley gave a start to former Swans defender Joe Walsh, 20, who was released from the Liberty Stadium in the summer after two League Cup appearances for the club.
The Welsh Under-21 international spent five years with the Swans after coming through the youth ranks.
Nathan Byre also made his debut after completing a loan move from Spurs yesterday. Claude Davis returned from injury, while Jon Forte, Billy Clarke and Dannie Bulman were also recalled to the side in place of Gary Alexander, Nicky Ajose and Mike Jones who was cup-tied.
P Jones, Byrne, Sadler, Walsh, Davis, Clarke, Akpan, Simpson, Adams, Bulman, Forte. Subs: Davies, Connolly, Torres, Alexander, Hunt, Ajose, Kuipers.
Tremmel, Tiendalli, Davies, Monk (capt), Ki, Britton, Gower, Michu, Richards, Routledge, Moore. Subs: Cornell, Tate, Situ, Lucas, Alfei, Donnelly, Graham.
3,963 (937 away)
Early exchanges were even before the Swans started to find their men with some neat inter-play.
Ki was clearly enjoying his new role in the heart of the Swans defence as he spread some great early passes out from the back, much to the delight of the 1,000 Jacks behind Gerhard Tremmel’s goal.
The first chance of the game didn’t arrive until the 13th minute though as Luke Moore collected a wayward Crawley pass to flash a fierce drive just over from 25 yards.
Josh Simpson drilled a 25-yarder inches wide of Tremmel’s post at the other end with 18 minutes gone, before Crawley’s Nathan Byrne became the first player in the referee’s notebook for a handball.
From the free-kick, Mark Gower’s in-swinger was met by the head of Michu inside the area, but his effort was easily saved by Paul Jones.
Swansea continued to pass the ball about and Michu put Wayne Routledge through in the 25th minute. The winger darted into the area, but his shot hit the side netting.
The visitors were growing in confidence as the half progressed and Jones had to race out of his goal to deny Michu reaching a Moore through ball.
There was no denying the Spaniard in the 26th minute, however. Tiendalli broke play up with a strong tackle before Ki and Gower moved the ball from the back. Routledge picked it up wide on the left and his pass inside was finished off superbly by Michu
from the edge of the area.
Six minutes later and Michu, with five goals to his name this season, was booked for a foul on Dannie Bulman.
The visitors were still in control though as Britton played in Moore who forced a good save from Jones before Michu headed a Tiendalli cross over the bar with seven minutes of the first-half remaining.
Two minutes from time the Swans were given a scare as Crawley applied late pressure and Billy Clarke was unlucky to see his goal-bound overhead kick blocked in a crowded area.
But as the game moved into injury-time, Crawley drew level as Josh Simpson’s shot from the edge of the area took a big deflection and flew beyond Tremmel for the equaliser.
Half-time: Crawley 1 Swans 1
An unchanged Swansea nearly took the lead for a second time four minutes after the restart when Moore met a Tiendalli cross, but Jones was equal to the task as he pushed his shot around the post for a corner.
Crawley were certainly not prepared to throw the tie away though and were well in the game as Jonathan Forte and Simpson tested Tremmel in quick succession.
Manager Michael Laudrup made his first change in the 58th minute as Rory Donnelly came on for his debut in place of Michu.
Three minutes later and Crawley were ahead for the first time as Hope Akpan turned Ben Davies inside the area to fire a low shot beyond Tremmel from 10 yards.
Laudrup responded by bringing on Danny Graham for Jazz Richards in the 67th minute.
But as the Swans pushed forward looking for the equaliser, they were nearly caught on the counter-attack as Forte forced a fine save from Tremmel.
It proved a costly miss for the home side as the Swans finally grabbed the equaliser in the 73rd minute when Tiendalli whipped in a great cross to the far post for Graham to head home.
Swansea were doing all the pressing as the game moved into the final stages, although Crawley still looked dangerous on the break as Forte forced Tremmel into a comfortable save.
The home side made two changes with five minutes remaining as Clarke and Forte were replaced by Nicky Ajose and Gary Alexander.
As the game moved into injury-time, the crowd started to prepare for extra-time. However, nobody had told Swans skipper Garry Monk as he met a Routledge corner at the far post to head home the winner.