After both sides remained dead-locked at the break, the visitors, thanks to goals from both Paul Armstrong and Allan Tait, were worthy winners on an afternoon where Boro looked totally out of sorts in midfield.
Indeed, the visitors should have taken the lead after only 42 seconds, with Charlie MacDonald slicing over the bar from close range.
Boro then missed two gilt-edged chances midway through the first half.
Firstly Chukki Eribenne was denied with a superb save from Phil Smith.
Then, Paul Harkness with time on his hands, fired past Smith only to be thwarted by the body Neil Jenkins on the line.
As the second half began MacDonald came close again for Crawley, driving fiercely from the right hand side on the tightest of angle as Mark Osborn fumbled on the near post in the 51st minute.
As the game seemed to coast along, the deadlock was broken a minute after the hour mark with a goal out of nothing for Crawley's Armstrong.
The midfielder, picking up a loose ball after a Boro clearance 30 yards out, volleyed home with a dipping lob that left Osborn with no chance.
Three minutes later the lead was doubled when Sacha Opinel found Tait with a long clearance from defence.
Tait, who was under pressure from Adam Theophanides, did superbly to keep his cool on the right hand side of the box before firing a low drive past the beaten Osborn.
The visitors' third came when Sean Hankin headed home Simon Wormull's corner in the 76th minute.
Boro did battle back late on with a header from 17 year old Michael Charles before Eribenne slammed home from the penalty spot after Opinel had handled in the box.
However, it was far too late for the home side.