But they did it the hard way, coming from behind to beat Carlton Town 2-1.
Poulter said: “Who knows how far we can go, it all depends who we get in the next round doesn’t it.
“If we get drawn at home we always fancy our chances.
“But we always travel in numbers away, so while it’s here and while it’s lasting we will certainly make the most of it.
“I said to the lads that our sole aim today was just to get through and we’ve done that. It’s a massive win for the club.”
Poulter said the Tigers knew it would be a hard tie, but thought their quality on the ball shone through.
“We didn’t see a lot of it really but when we did we made it count,” he said.
“Our front three were very good in the first half but kind of ran out of steam in the second.”
The visitors made a fine start, snatching the lead after five minutes when a Kerian Walker long throw caused chaos in the box to allow Edward Clarke to score.
The Tigers fought back and after defender Brad Wilson shot over from 20 yards the home side won a penalty.
Ross Goodwin went down under a challenge and Kyle Jordan levelled from the spot.
On the half-hour Steve Woolley curled in the winner with a 23-yard free-kick that beat the keeper at the near post.
In the second half Matt Sykes caused the visitors problems with quick feet and low crosses.
At the back, young full backs Brad Wilson and Liam Royles, making their first starts of the season, did well, while Matt Reay and Max Pemberton were commanding at the heart of the defence.
Poulter added: “Jamie Price failed a fitness test before the game so Brad kind of knew he was playing.
“Liam Royles has taken his chance in Michael Trench’s absence. I think they’ve given the gaffer a huge headache now after those displays.
“That’s the reason we have 20 players, each one of them can play and do well, so the players that aren’t here will ultimately have to win their place back. That’s part and parcel of football.”