Rabs came from a goal behind to beat their Conference hosts with a 90th minute Craig McPherson winner, securing a second round trip to face Aberdeen side Glentanar.
Maxwell had warned beforehand that Greenock was a difficult place to go, despite the Renfrewshire side operating a tier below Rob Roy – and says that’s exactly how it turned out.
He said: “The dynamics of the park are tight so you get into the box quite quickly, so there's a lot of play in the boxes but maybe not clearcut chances.
"We set up playing a diamond to try to narrow the game a wee bit and played really well but made one mistake - the goalkeeper never came for a cross and the boy got in at the back post to score with a header against the run of play
"Just prior on that we were one on one with their keeper and then the ball gets transferred up the park and they score.
"It was disappointing but we got an equaliser, which was deserved because we were the better team, and the they hit the woodwork and we finished the half strong.
"In the second half we started to create chances but just couldn’t get a goal. We hit the bar and got one cleared off the line while they never really threatened.
"And right at the death Craig McPherson scores a great header at the back post.
"In cup football there's no better time to score than in the 90th minute and I was delighted for the supporters.
"The cup gives them a break and gives them a few journeys and they turned up in their numbers yesterday.
Next up for is Saturday’s derby at home to Rossvale – and with midweek games now finished Maxwell is glad he will have extra preparation time in the weeks ahead.
He said: “It gives us a chance to settle down a bit and work in training, but every team will be the same. Every team will want a wee break from midweek games.
"I had seven players out on Saturday so it gives us a chance to reset.
"Rossvale have won a couple of games recently so it will be a tough game, but you don't get any easy games in our league."