Enzio Boldewijn gave them the lead in the 33rd minute with his fourth goal of the season.
He pounced from short-range following an initial shot by Mark Randall which was spilled by keeper Joe Fryer.
But Crawley had to play more than half the match with only ten players as just after their goal, Josh Payne was sent-off for a dangerous challenge.
Reds made six changes to the side lost 2-0 against Colchester on Boxing Day.
Goalkeeper Glenn Morris, Dannie Bulman, Enzio Boldweijn, Josh Payne, Mark Randall all returned while defender Josh Doherty made his first league start since September 12.
Yusuf Mersin, Cedric Evina, Moussa Sanoh, Thomas Verheydt and Billy Clifford were all dropped to the bench while Kaby Djalo was not included in the squad.
Stevenage made two changes with Ben Kennedy and Fraser Franks coming in for Danny Newton and Terence Vancooten.
In the early exchanges Jordan Roberts had a go from the edge of the box which flew wide.
Then Boldewijn went on a fantastic run from his own half, beating several opponents before shooting straight at Fryer.
Crawley took the lead when Randall’s shot from outside the area was saved by keeper Fryer and Boldewijn netted on the rebound.
However joy turned to anger among Reds fans when just three minutes later they were reduced to ten men when Payne was sent-off for a dangerous tackle on Alex Samuel.
Stevenage’s Joe Martin forced a save by Morris with a turn and shot.
The visitors didn’t take advantage of having an extra man during the whole of the second half and although captain Mark Connolly led the defence well, they were not under that much pressure.
In the closing stages Morris showed his great value to the side by making an amazing save to parry a shot by Matt Godden over the bar.
Crawley Town: Morris, Young, Roberts, Bulman, Boldewijn, Lelan, Payne (Meite 82), Randall (McNerney 61), Doherty, Connolly (capt), Yorwerth
Unused subs: Mersin, Evina, Sanoh, Verheydt, Clifford
Stevenage: Fryer, J.Smith, Martin, King, Franks, Kennedy, Samuel (Gray 75), Gorman (Whelpdale 60), Godden, Henry (capt) (Pett 60), McKee
Unused subs: Wilkinson, Day, Conlon, Wilmott
Referee: John Busby
Attendance: 2,075 (224)
Man of the Match: Josh Lelan