Torquay, three points adrift at the bottom of League Two, out fought promotion-chasing Notts County in heavy conditions seven days after the two sides had drawn 2-2 at Meadow Lane in the league.
It was Bedeau's finishing which did the trick, earning Torquay's first regulation win in 11 games - the first round victory over Harrogate came via a penalty shoot-out.
The Gulls even had to overcome the early loss of on-loan centre forward Ian Stonebridge, who limped off with an ankle injury after only 11 minutes.
Torquay were already injury hit and it forced head coach Leroy Rosenior to play a makeshift pair up front of substitute Kevin Hill and Bedeau, who has played most of his career on the right wing.
They still dominated the first half with the wind behind them but had only Bedeau's first goal in the 45th minute to show for it.
He squandered an inviting header in the 39th minute but he could hardly miss on the stroke of half time.
County keeper Kevin Pilkington parried his first header and when Adam Lockwood rolled the rebound across the goal, Bedeau had only to tap the ball into the empty net.
Pilkington kept County in the match with a string of brilliant saves early in the second half.
Lewis McMahon's 54th minute equaliser - a 22-yard right-foot shot - came against the run of play.
But after Glynn Hurst missed a free header, Torquay responded well in heavy conditions and deservedly won it in the 69th minute.
Matt Hockley supplied the through ball and Bedeau raced clear on the right to chip a 20-yard angled shot over the advancing Pilkington.