In a six-goal thriller Bournemouth came from 2-0 down to level the scores at 2-2 with 20 minutes to play but Fallon and Peacock's goals were enough to seal the three points for Town.
Jamie Cureton gave the Robins a 37th minute lead when he latched on to Stefani Miglioranzi's cross before turning well in the area and delivering a low right-footed finish.
It was Cureton's first goal for the club since signing in August and came against one of his former teams where he spent a period on loan at as a teenager.
Miglioranzi then helped himself to his first of the season when he capitalised on Gareth Stewart's parry from Cureton's effort to fire the rebound high into the net.
Swindon then seemingly went to sleep as Bournemouth pulled themselves back into the game. James Hayter crossed for Dani Rodrigues to volley home. Hayter then got himself on the score-sheet with a well-taken penalty on 66 minutes for his 15th goal of the season.
Swindon though, somewhat against the run of play, sealed all three points when Fallon headed home his 14th goal of the campaign.
Right-back Steve Jenkins chased a lost cause down the right-hand channel and flung the ball into the penalty area for Fallon to direct the ball into the bottom corner.
Paul Smith almost wrapped things up on 83 minutes when he was unlucky to see Stewart save his dipping volley.
But Peacock, on as a 75th minute substitute, atoned for an earlier miss by putting the scoreline beyond doubt at the death.
The former Sheffield Wednesday striker robbed Stewart as he rushed a clearance, held off the attentions of Neil Young and then side-footed into an empty net to the delight of the ecstatic home supporters.