PNE took the lead through Andrew Hughes as he headed home a Ben Whiteman corner and fellow Joe Rafferty made it two before the break.
Cheltenham got themselves back in it after the break, Kyle Vassell vacating the bench to blast his side back into the game, but goals from Sean Maguire and Emil Riis put the game out of reach.
The game was not without casualties however, with Connor Wickham and Liam Lindsay both being forced off before the interval.
McAvoy felt his men coped well with the adjustments that had to be made and was pleased with the quality of their football.
"I thought we were excellent," he said.
"Some of our play, particularly in the first half was very very good. 2-0, we fully deserved that.
"I was a bit concerned with Connor's injury early on, we worked on things yesterday and then you lose one of your players straight away. Sean has been great for us anyway, then he comes on and it doesn't disrupt us.
"Then Liam getting the injury, it knocks you a little bit because we need to move Jordan Storey into the middle of the centre back and bring on young Sepp van den Berg. Thankfully it didn't hinder us.
"I did get a fright when there was a tackle on Sean after 30 odd minutes and I'm thinking 'surely I'm not going to have had to make three subs within the first 30 minutes of the game'.
"Thankfully it didn't impact on our play and some of it was really good."
The visitors had a spell during the 90 that could have seen them find a leveller, North End had lost their stranglehold on the contest and the Robins were growing in their threat levels.
The PNE head coach was pleased with how his side saw off the uprising and backs the spirit in the camp to see them through - which is bouyed by the receptions they receive from the supporters.
McAvoy said: "I thought the fans tonight were great, I could feel it myself when I was coming out. I thought they were up for it.
"At 2-0, I said this at half time, it's a precarious scoreline. The next goal can decide how the game goes, as we found out in the last round against Morecambe as they scored just before half time, then it becomes a battle.
"I knew if we got the third goal it'd be an easier night for us. Credit to Cheltenham, they came out and pressed us really hard in the second half - although I thought we started well.
"Some of the football, particularly on the left hand side, was excellent.
"Goals change games. It was a great goal from Vassell from their point of view and then it's game on.
"This group has a good team spirit and we knew what the task was tonight, go and win the game.
"It didn't matter how we did it, just go and win it. They've stood up to it tonight again."
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