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The Mariners took the lead through Martyn Woolford on 14 minutes and held their advantage at the break despite coming under pressure from their hosts.
Their task was made tougher when Alex Whitmore was shown a straight red card just six minutes into the second half and the Exeter pressure eventually told when Jayden Stockley headed them level with 12 minutes to go.
But the Mariners recovered and grabbed their lead back within two minutes as Mitch Rose came off the bench after a penalty had been given to score from the spot with his first touch.
Town then had to survive attack after attack but a superb display from goalkeeper James McKeown and his defence ensured they held firm to earn all three points and keep their excellent run of form going.