Cox bundled the ball home from close range inside 10 minutes after the Boro defence failed to deal with Leon McSweeney's cross.
Orient continued to pile on the pressure, with on-loan debutant Shaun Batt linking well with Charlie MacDonald and both going close to doubling the lead.
Stevenage had looked rusty in their first match for three weeks but soon picked up, Sam Hoskins producing a flying save from O's keeper Jamie Jones after a terrific run.
Jones had to palm the ball off the line after a goalmouth scramble from the resulting corner, while Felipe Morais also went close for the hosts.
The visitors started the second half as they did the first, MacDonald heading over from close range and later scoring from an offside position, while substitute David Mooney sent a 20-yard strike over.
Boro regained possession as the tempo dropped towards the end but did not trouble Jones.