Substitute Isaac Olaofe struck for the only goal of the match in the 84th minute after Tom Nicholls saw his penalty saved earlier in the second half for the Red Devils.
Sutton stretched their winning streak to four matches on the trot and are now fifth in the table, just five points off of the top spot and two points from an automatic promotion place.
Story of the match
Crawley dominated the early stages of the game as Nichols forced a save out of Sutton goalkeeper Dean Bouzanis at his near post. At the other end, Rob Milsom's free-kick was tipped over the bar by Glenn Morris.
On 33 minutes, Richie Bennett saw the ball fall into his path some thirty yards out and he unleashed a ferocious shot which was matched by the quality of the save from Morris.
Donovan Wilson almost had a goal to his name before the second half was a minute old as Bennett's flick sent Enzio Boldewijn in down the left, and his low cross was met by Wilson with a first time effort deflected just wide by Tony Craig.
On the hour mark, George Franconmb took the ball inside his own half and began driving forward. After playing a one-two with Nichols, the Crawley captain found Will Ferry with a perfectly weighted through ball, but the young winger couldn’t muster up enough power to trouble Bouzanis.
Moments after coming on, Olaofe got past his marker and his volley was deflected out for a corner. Boldewijn saw his shot saved by Morris and Harry Beautyman sent his chance high over the net.
Boldewijn gave away possession on the halfway line and it almost resulted in the opening goal. A slick passing move sent Nicholls free on the right and as Nick Tsaroulla arrived to meet the resulting cross and David Ajiboye ran into the back of him.
Referee Tom Nield pointed to the spot, and after Louis John and Milsom had both been booked for their protests, Bouzanis dove to his right to keep out Nicholls' penalty.
Finally, the Yellow Army grabbed the winner. Milsom's free-kick was headed back by Joe Kizzi but looked to be a comfortable claim for Morris until Nicholls' wild swipe sent the ball high into the air, and Olaofe managed to head home.
Red Devils substitute Ludwig Francillete nearly grabbed an equalizer in injury time when his looping header forced Bouzanis to stretch and tip the ball over. Sutton held on to move into the top five.
Man of the match: Dean Bouzanis
He made some key saves, including one in the first half and also for Nicholls' penalty as he recorded his fifth clean sheet of the season and is part of a Sutton defence that has allowed just three goals during the Yellow Army's current winning streak.