Luke Guttridge’s spectacular volley booked Northampton Town a spot in the League Two play-off final.
Cobblers goalkeeper Lee Nicholls saved Marlon Pack’s 15th-minute penalty after Ben Tozer was adjudged to have fouled Russell Penn in the area.
Northampton, 1-0 up from the first leg, doubled their advantage on 27 minutes through Guttridge’s 20-yard strike.
Kaid Mohamed’s header hit the bar for Cheltenham but the visitors held on to earn a Wembley date with Bradford City.
The Bantams had already overturned a first-leg deficit in defeating Burton Albion in the other League Two play-off semi-final earlier in the day and Cheltenham were hoping to do something similar at Whaddon Road.
Northampton’s first-leg hero Roy O’Donovan had the first genuine sight of goal in the second leg, but this time he was less clinical. Having latched onto Adebayo Akinfenwa’s accurate pass he was unable to slot the ball beyond goalkeeper Scott Brown, who got the slightest of touches to turn his shot away.
The Robins, unbeaten in 11 matches on their own ground coming into the play-offs, continued to have the better of the general play though and appeared fortunate to be handed a golden opportunity to level the tie.
Tozer’s challenge on Penn was reckless and referee Andrew Madley did not hesitate in pointing to the spot, despite replays suggesting that the contact had come outside the penalty area.
The Cobblers would have felt justice was done, however, when Nicholls produced a brilliant one-handed stop to keep out Pack’s effort.
Nicholls, on a season-long loan from Wigan, was again Pack’s nemesis soon after, this time moving smartly to his right to keep out the 22-year-old’s 25-yard free-kick.
With the game continuing to seesaw from one end to another, where Pack had failed from distance at one end Guttridge spectacularly succeeded. His dipping volley gave Brown no chance and with it Northampton took a giant step towards Wembley.
While the first period had been an open affair, albeit with both teams looking most dangerous from set-pieces, the second was scrappier and, aided with a two-goal cushion on aggregate, the visitors looked to be relatively comfortable.
In an attempt to alter the momentum of the tie, Robins manager Mark Yates turned to joint-second top-scorer Mohamed and the substitution almost paid off with immediate effect.
Right-back Sido Jombati got to the byline to deliver a superb cross for his former Bath City team-mate Mohamed and, although Nicholls was at last beaten, his thumping header came crashing off the crossbar and away.
Sensing their opportunity, the home side continued to press and Sam Deering stung the fingertips of Nicholls from close range when he might have done better.
Shaun Harrad was denied by the 20-year-old goalkeeper once more from close range as Cheltenham piled bodies forward, but Northampton played out the closing stages to secure a trip to Wembley.