Late goals from Chris Carruthers and Jefferson Louis helped Crawley end their alarming run of just one point from a possible 12.
The game looked destined to end in a goalless draw, the hosts in particular spurning a number of chances.
But on-loan Carruthers finally broke the deadlock with three minutes left.
Louis then poured salt into Histon's wounds with a second in injury time to end Crawley's four-game run without a win and move them into mid-table.
The Red Devils dominated the opening stages, but Eddie Hutchinson, Thomas Pinault and Charles Ademeno, who had another effort parried away by Danny Naisbitt, all missed the target.
Histon keeper Naisbitt then made another fine save to deny Sam Rents, before the hosts wasted yet another opportunity when Kieran Djilali dragged wide from close range.
Histon's Michael Frew forced Crawley keeper Simon Rayner into his first save, but Crawley remained the dominant force, Djilali going close again shortly before half-time.
Djilali saw another effort tipped around the post at the start of the second half, while Craig Pope headed wide for the visitors from 12 yards.
Pope and Josh Simpson also brought saves from Rayner as Histon started to grow in confidence. And Nathaniel Knight-Percival wasted a glorious chance to put the Stutes ahead in the 70th minute when he missed the target with a header from Gareth Gwillim's corner.
Crawley finally took the lead on 87 minutes when left-back Carruthers, on loan from Oxford, latched on to Djilali's pass before firing an unstoppable shot into the top right corner.
And the hosts added a second at the death as the outstanding Djilali set up journeyman striker Louis to slot home from 12 yards.
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