Accies beat League Two side Arbroath 2-1 in the League Cup first round with goals from Dougie Imrie and Louis Longridge, but Neil accepts Inverness will present a much tougher assignment on Saturday.
He said: "We are through to the next round of the Cup but we know we are going to have to pick up the pace of our game for next week. You find that good players tend to rise to the occasion and I'm hoping that will be the case next week.
"I thought we totally controlled the match. They got a goal against the run of play once we had brought some of the younger boys on.
"I was a bit disappointed with the last five minutes but we controlled the game and had a lot of the ball. It was just a case of breaking down a team that were well-organised and sat in and made it as difficult as they could."
Arbroath boss Allan Moore added: "We have played opposition who are a couple of leagues above us and we knew it would be difficult coming here.
"I'm disappointed that we went a goal down so early in the game but I'm proud of the boys for the way they stuck by the task."