The League One newcomers looked impressive throughout at the Den and secured victory after Jones saved penalties from Darius Henderson and Jack Smith in recently-appointed manager Richie Barker's first match at the helm.
Things began well for Crawley, who deservedly broke the deadlock after 16 minutes when Nicky Adams' floated corner was headed home by Hope Akpan.
The lead lasted just seven minutes, though, as Liam Feeney found space to twist and turn on the right flank, before delivering a low cross that Darren Ward nodded past goalkeeper Jones.
Play was flowing from end to end as both sides failed to control possession and saw attacks thwarted by last-ditch challenges or wayward finishes.
It took a moment of magic for the deadlock to be broken once more and it came from the boot of Adams, who curled a superb shot from distance into the top-left hand corner over the statuesque Maik Taylor.
Former Millwall striker Gary Alexander saw a goal chalked off as Crawley continued to press, although the Lions managed to eke out a leveller and force extra time with five minutes remaining through substitute Shaun Batt.
Neither side were able to find a way through in the added 30 minutes - Henderson and John Akinde both missing gilt-edged chances - so the match went to penalties.
Henderson, who also saw a goal disallowed in extra time, had his penalty saved, before Smith was then denied by Jones as Crawley went on to win it 4-1.