With the game only three minutes old, the visitors took the lead when Darren Polehill received a superb through ball and broke through to score with a fierce shot.
Ten minutes later Minehead found themselves 2-0 down when Theo Shell scored with a tremendous shot from the edge of the box.
Fighting their way back into the game, Minehead went close to reducing the arrears when Nicky Woon broke clear of the defence only for the keeper to come out quickly and block his shot, and Paul Meare put the rebound over the bar.
At the beginning of the second half Minehead took command of the game for a period and in the 53rd minute leading scorer Woon gave them some hope when he fastened onto a loose ball to score.
Nine minutes later, however, Bideford restored their two-goal lead when Shell turned a low cross into the net.
Five minutes later the game was put out of Minehead's reach when Darren Knowling brought Shell down in the penalty area and Adam Shearer scored from the spot to make it 4-1.
Despite being outplayed, Minehead stuck valiantly to their task and the injured Gareth Morgan came on in the 76th minute as substitute for Shane Porter.
With three minutes remaining, Lee Porter pulled a goal back for the home side, but two minutes later Bideford rightly had the last say when Richard Coomm, who had an outstanding game, scored from another low cross to make the final score 2-5 and keep Minehead in trouble at the bottom end of the league.
Minehead: L Ayres, D Burr, A Reader, A Giblett, D Knowling, L Porter, P Meare, S Nicholls, N Woon, D Palfrey, S Porter; subs, N Wyborn, G Morgan (both used).