After a goalless first half, Jabo Ibehre struck for the first time in seven games to give Colchester the lead on 62 minutes with a fine header.
And Ibehre made the game safe after 73 minutes when he tapped in following a fine run from Gavin Massey.
"We didn't look like a relegation team. We can't worry about the pressure, I told them I'll take the bullets, they just have to play," said Dunne.
"We've been in and out of the bottom four, we deal with this pressure every day, it would be nice to know what it's like to be safe and not have the pressure but we're used to it. We work every day under it.
"We can't let our standards drop at all, and we showed we didn't. We know that if we do we'll get beaten. We don't do draws, you have to have that level of commitment and aspiration to stay in the league.
"I felt all we needed in the second half was to better those finishing touches and we got them."