Andrew Hughes, Joe Rafferty, Sean Maguire and Emil Riis found the net for the Lilywhites but the win did come at a cost with Connor Wickham and Liam Lindsay both forced off injured in the first half.
Hughes and Rafferty were on target for the first time since 2019 and North End were good value for their 2-0 half-time lead.
Kyle Vassell pulled one back for the visitors early in the second half and that led to something of an indifferent spell from the home side who lost a bit of control of the play.
However, they rallied in the latter stages and substitute Maguire - on since the seventh minute - struck the third goal in the 82nd minute.
The best goal of the night came from Riis at the end of stoppage-time, the Dane driving past a couple of defenders before drilling home a fine low finish from the edge of the box.
Victory extended PNE's unbeaten run in all competitions to seven games.
Frankie McAvoy had made seven changes to the starting XI as he went for a spot of squad rotation.
But inside the opening seven minutes one of those he gave a start to, striker Wickham, was forced to leave the pitch after a collision in the opening exchanges left him limping.
That brought Maguire into the action a lot sooner than he might have expected.
Bar a deflected shot from Cheltenham's Taylor Perry which went straight at Declan Rudd, North End bossed the first half.
Skipper Greg Cunningham fired a first-time shot over the bar after Riis' long throw-in had been cleared to the edge of the box.
Rafferty saw a low shot parried by Robins keeper Owen Evans, PNE keeping the ball alive in and around the box in the aftermath to eventually force a corner.
It was from that flag kick that they took the lead, Ben Whiteman delivering it into the middle where Hughes met it with a bullet header from six yards.
The visitors' Dylan Barkers was booked for a late challenge on Liam Lindsay two minutes after the goal, a tackle which on another night could have merited a red card.,
Lindsay played on for a few minutes after treatment before becoming the second player to see his night end early.
North End doubled their advantage in the 37th minute, Cunningham dropping a fine cross into the box from the left channel.
Rafferty met it in the box with a cushioned shot which looked on target before getting the last touch off a Cheltenham player.
In the run-up to the interval, Cunningham was played into space just inside the box by Maguire's pass, the left wing-back's shot saved by Evan diving to his right.
Whiteman saw a shot from 20 yards deflect just wide, with Ali McCann off target with a low drive.
Buoyed by his goal, Rafferty had a shot two minutes into the second half, his effort from outside the box saved by the keeper.
Cheltenham halved the deficit in the 55th minute, Vassell making an immediate impact after coming off the bench.
Vassell had been on the pitch barely two minutes when he lashed a shot from the corner of the six-yard box into the roof of the net, Conor Thomas having headed a cross into his path.
That goal led to a ragged period of play from North End who seemed to be knocked out their stride by it.
Whiteman sent a shot wide, while Vassell volleyed over Declan Rudd's bar at the other end.
In the 68th minute, Maguire linked-up with Riis and got a shot away which Evans dived to push past the post.
Maguire had looked a threat all night since his early introduction and deservedly found the net for PNE's third goal in the 82nd minute.
Rafferty's low cross from the right-wing found the Irishman to the right of goal, near to the corner of the six-yard box.
Maguire took a touch, turned and fired a low right-foot shot across the keeper into the far corner.
There was almost a quick lifeline thrown the visitors' way after the goal, Vassell taking the ball round Rudd but Rafferty forced him wide and blocked his shot on the byline to the left of goal.
But it was PNE who had the last word, Riis taking the ball past a couple of defenders before driving a low right-foot shot into the far corner from the edge of the box - The Dane's seventh goal of the season and fourth in this competition.
PNE: Rudd, Storey, Lindsay (Van den Berg 31), Hughes, Rafferty, Whiteman, McCann, Cunningham, Potts (Browne 80), Riis, Wickham (Maguire 7). Subs (not used): O'Reilly, Earl, Rodwell-Grant, Iversen.
Cheltenham: Evans, Horton, Pollock, Freestone, Blair, (Ponds 77) Barkers, Thomas (Raglan 77), Chapman, Perry, May, Williams (Vassell 53). Subs (not used): Long, Armitage, Flinders.
Referee: Dean Whitestone (Northamptonshire)
Attendance: 6,561 (218 away)
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