The Robins brushed aside their more experienced opponents with three first-half goals and also hit the woodwork on two occasions.
Boston created chances but it seemed that nothing would go right for them in front of goal. Their misery was compounded three minutes from time when defender Ben Futcher was given a red card by referee Neil Swarbrick for a second bookable offence.
Boston had actually created the first chance of the game when Cheltenham goalkeeper Shane Higgs was forced to save sharply from David Noble, but the Cheltenham front two of Damian Spencer and Kayode Odejayi were proving a very big handful for the visitors.
Tricky wide man JJ Melligan was also causing damage down the right and all three players combined to open the scoring on 19 minutes.
Odejayi broke away on a strong run down the middle of the pitch and passed wide to Melligan, whose curling cross was touched in by Spencer from six yards.
Midfielder Grant McCann smacked the crossbar with a 35-yard free-kick midway through the first half but it was during the five minutes leading up to the break that Cheltenham took control.
First, Jamie Victory sent a looping header from 15 yards into the net from a McCann cross, then Odejayi latched onto a pass from Jerry Gill and burrowed across the box from the right.
His low centre appeared to be heading across the face of goal until it deflected off Futcher and crept inside the far post.
Craig Armstrong almost made it 4-0 when he headed against the post from a Melligan cross and Cheltenham also had strong penalty claims when Spencer went to ground under a challenge from goalkeeper Nathan Abbey.
The home side took their foot off the gas in the second half and Boston had chances to score, although goalkeeper Shane Higgs made fine saves from both Jason Lee and Julian Joachim.
Futcher received his marching orders for a foul on Odejayi after earlier being booked for dissent, and Cheltenham substitute Ashley Vincent went close to making it a quartet of goals when blasted wide after a strong run through the middle.