Oldham’s young goalkeeper Joel Coleman had to go full length to push a goal-bound shot from Dean Morgan round the post before McLeod gave Crawley the lead in the 15th minute.
The skipper rose to expertly head a cross from Gavin Tomlin into the top right-hand corner for his 20th goal of the season.
Central defender Wood, on loan from Rotherham, struck to double the advantage in the 34th minute by shooting home from close range after Lee Fowler had driven a free-kick through the wall.
Former Crawley midfielder Mike Jones produced Oldham’s best moment of the first half with a clever free-kick, which alert goalkeeper Brian Jensen pushed round the post.
Jensen superbly tipped over a rasping shot from Rhys Turner shortly after the interval before Crawley midfielder Gavin Tomlin wasted a good opening after being set up by McLeod.
Oldham struggled to impose themselves on the game and Jones had a 25-yard drive saved by Jensen midway through the second half.