To my eyes, it wasn't really edited to pieces -- the only three edits were the three close-ups of Bob pressing a button and the camera following the chase lights up to a flap opening. And those were simply silent video clips placed under the audio from the original playing.
If you listen to the audio alone without watching the video, it all makes sense. We know the chase light sound effect is being played manually. So when Bob pressed the first button, the sound effects operator played the sound effect. Upon seeing that the lights weren't working, they hit the buzzer at the end of the sound effect anyway -- they had already been signaled that $2.80 was wrong and the show must go on. Bob, expecting the flap to open, repeated "I believe" as the sound effects operator pressed the buzzer again to let him know that despite the flap not opening, that choice was incorrect. Bob heard the buzzer and repeated "You're wrong" to establish that even though nothing happened, the eggs weren't $2.80.
At that point, everybody on the staff and crew understood that they'd be playing the rest of the game with a broken board -- they knew what the right answers were; they could finish without the flaps opening. Bob continued to go through the motions of pressing buttons, and the sound effects operator just played the appropriate sound effect after the chase noise finished. Note the slight delay of Bob confirming the two correct answers -- it doesn't make sense if you watch the video, but if you close your eyes and listen for the sound effect like everybody in the studio was doing, the timing feels completely natural.
Either after the show went to commercial or possibly after the credits rolled, the crew got the board working properly and then they shot the pickups of Bob pressing each button successfully. I agree that the edits sync pretty well (although they needed a little extra bit of footage of Bob's hand just hanging out next to the $2.80 button before pressing it in order to cover the live-in-studio shot).
