Watt, on loan from Arsenal, struck just two minutes after Boro had been reduced to 10 men thanks to Anthony Grant's second yellow card.
Boro goalkeeper Steve Arnold was called into action early on as he foiled Gavin Massey on four minutes, while Colchester keeper Mark Cousins pulled off a spectacular diving save to turn over Filipe Morais's superb shot midway through the half.
The game changed six minutes into the second half when Stevenage midfielder Grant received a second booking for a foul on Watt and was sent off.
Colchester capitalised on their one-man advantage by putting the ball in Boro's net three minutes later - Watt running at the Stevenage defence before unleashing an excellent low, left-foot shot to bury the ball in the far corner.
Arnold then denied Massey and Ibehre to keep the score at 1-0, before Anthony Wordsworth clipped the bar with a free-kick for the hosts. Stevenage went close to grabbing an equaliser at the death, but Bondz N'Gala's effort struck the bar, allowing Colchester to claim victory.