Bideford grabbed an equaliser six minutes into injury-time as Melksham had to settle for a 1-1 draw on their travels.
Melksham defended manfully throughout and Nat Auckland put them ahead with a neat finish.
But just when it looked like Wayne Thorne's men were set to hang on for a battling away win, Bideford equalised with the last kick of the game.
Town assistant manager Dave Clayton said: "We defended well for most of the game, like you have to against Bideford, and we took our chance when it came.
"It was sickening to let in a goal so late. I don't know where the referee got six minutes of injury-time from because there weren't any injuries in the second half."
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