With the score level at 2-2 and extra-time beckoning in Tuesday night's Capital One Cup third-round tie at the Broadfield Stadium, Monk nodded home a corner by Wayne Routledge in the 90th minute as Swansea evaded a third successive defeat.
The Barclays Premier League club looked on course for an easy night as their early dominance was rewarded when Miguel Michu pounced.
But Josh Simpson and Hope Akpan struck to give Crawley, who lost to Manchester United in the fifth round of the FA Cup two seasons ago, hope of humbling their top-flight opponents.
Had the finishing skills of Jonathan Forte matched his surging runs from midfield, a prized scalp would have been assured.
However, substitute Danny Graham equalised in the 76th minute, sparking a lively spell from Swansea who finally regained the lead through Monk.
It took 12 minutes for the first goal to come under meaningful attack with a defensive error by Crawley right-back Nathan Byrne gifting the ball to Luke Moore.
Moore dashed to the edge of the area where he pulled the trigger only for his shot to whistle past the right post.
An almost identical error of judgement saw Byrne cause panic among the home defence for a second time, but this time Moore was hustled off the ball.
Michu nodded a free-kick straight at keeper Paul Jones before Routledge stabbed a glorious chance into the side netting.
But Routledge made amends in the 27th minute by unlocking the Crawley defence with a mazy run that set up the opener for Michu.
Stationed on the edge of the area, Michu steered the ball over the head of Jones with a superb finish to grab his fifth goal since his summer arrival from Rayo Vallecano.
The Spanish attacking midfielder was then booked for fouling Dannie Bulman and later sent a header over the crossbar, but only after More had squandered another great opportunity.
The closing moments of the first half saw Swansea come under pressure and they were forced to deal with a dangerous overhead kick from Billy Clarke.
Worse was to come, however, as Crawley made their breakthrough when Simpson gathered a failed clearance from 20 yards out and drilled home a heavily-deflected shot.
Moore saw a promising attempt bundled out for a corner by Jones, but it was Swansea who were cracking in the early stages of the second half.
Forte surged through midfield and tested keeper Gerhard Tremmel from 20 yards out, before Nicky Adams failed to capitalise on the follow up a heartbeat later.
Forte was also involved as Swansea's worst fears were realised, Akpan taking his incisive pass, turning and slotting home from close range.
Having run half the pitch, Forte should have put Crawley 3-1 up but his shot was easily saved by Tremmel.
The home side were left to rue the missed chance as in the 74th minute Swansea equalised when Graham nodded in Dwight Tiendalli's cross.
And the decisive blow was struck as the match entered injury-time, Monk jumping to head home Routledge's corner.
|Team Name||Crawley Town||Swansea City|
|Shots On Target||7||7|
|Shots Off Target||8||6|