Goals from Adam Stansfield and Liam Sercombe put Exeter in front before Tom Craddock pulled a goal back for the Hatters, but Harford was unhappy that his side were unable to unlock Exeter's resilient defence in the second half.
Harford, whose side now trail 22nd-placed Grimsby by 16 points, said: "They had a lot of possession and we showed them a bit too much respect but even so we created six or seven good chances and hit the crossbar twice.
"We controlled the game for the second 45 minutes but to be fair to Exeter they got bodies behind the ball and defended deep and we never really tested their goalkeeper.
"It's disappointing because we get ourselves into a position where we feel we can achieve something but then we falter.
"We'll never give up though and I believe we can still stay up."