Bedeau volleyed home in the 65th minute to dent Peterborough's play-off ambitions and clinch Torquay's first win in six games.
With new football advisor Ian Atkins helping manager John Cornforth on the touchlines, the Gulls took control from the start and never relinquished it.
And on the pitch it was new loan signing Danny Hollands from Chelsea who sparked the home side's midfield.
Hollands, 20, needed only a goal to cap an impressive debut, and he nearly got that, too, with an outrageous 40-yard dipping volley which flew just wide on the stroke of half-time.
When Posh did finally wake up, left winger Jamie Day forced Steve Woods to head off the Torquay goal line in the 28th minute. But that was one of only a handful of serious attempts on goal from the visitors.
In the 65th minute Martin Phillips crossed from the right wing, Bedeau controlled the ball and hit a right-foot volley in off the far-post from 12 yards.
The hard-working Paul Robinson headed just wide from a Bedeau cross with 15 minutes left.
In a tense last quarter of an hour, one Danny Crow cross went begging for Posh and Paul Carden also shot wide from a good position.
But, with right-back Matt Villis outstanding in a solid defence, Torquay seldom looked like losing their grip.
With most other results going in their favour, the result proved even more important for the Gulls, lifting them off the bottom of the table and allowing them to narrow the gap on the group just outside the relegation zone.
And it also extended a miserable run for Peterborough at Plainmoor. They have now won only once in their last 15 trips to South Devon.