Early on, Lewis Putman's free kick was turned aside by Kyle Moore, whilst Slough keeper Mark Scott blocked well from Grant after two crunching tackles in midfield led to the chance opening up for the striker.
Slough piled the pressure on the Bideford goal with a series of corners, but struggled to test Moore. Adrian Sear headed one effort wide.
Then, on 29 minutes and against the run of play, Josh Grant found himself with time and space to slot the ball past Scott as Slough's defensive frailties were apparent again.
Slough had a let-off ten minutes later as the visitors hit the bar with a header from Ben Wood's cross.
A minute before the interval, Sear teed up Adam Martin for a shot, but it was deflected narrowly over the bar. Bideford then broke away and Grant was left in a good position to threaten Scott's goal, but he miscued horrible and the ball went well wide.
Mpi's two goals, on 59 and 61 minutes, came at an excellent time for the Rebels. First Mpi slammed home a loose ball from six yards after a long throw wasn't cleared by the visitors.
Then, with Bideford still reeling, Mpi sidefooted home from a few yards out after a header across goal from Gary MacDonald.
After Billy Tucker went on a marauding run down the right which ended in a shot being deflected into the side netting, Mpi went close to a hat trick, but hit the bar on 71 minutes.
A few minutes later, Slough survived a goalmouth scramble after a corner, but eventually cleared their lines.
Bideford gave Slough a few nervy moments in the closing stages, but Slough saw the game out as Bideford ended the game with nine men after two dismissals.
The visitors looked to have an excellent claim for a penalty after a trip from Guy Hollis, but the referee waved the appeals away and Ben Gerring was shown a second yellow card for dissent. Ian Sampson joined him two minutes later after a poor challenge on Scott.
Bideford Line up
Kyle Moore, Ian Sampson, Stephen Reed, Billy Tucker, Ed Palmer, Ben Gerring, Matty Bye (c), Matt Hockley, Josh Grant, Matty Andrew, Ben Wood. Subs: Kevin Squire, Dan Harrison, Chris McGrath.