The U's struck two goals inside six minutes in the second half against a distinctly lacklustre Carlisle and earned themselves a standing ovation from the home fans.
Gavin Massey forced in the opener and defender Tom Eastman's 71st-minute far post header proved the clincher for the U's who had to get at least a draw to stay up.
Dunne said: "We have had our ups and downs, but we kept believing and we have stuck together. The planning starts here and we move on.
"Commiserations to Scunthorpe. It could have been us. We could have been in League Two, but the club is going in the right direction.
"It's mission accomplished and we got the job done. I thought we were magnificent. We had a game-plan and got it spot on. We frustrated Carlisle and I said at half-time we could go for it and win the game.
"I'm pleased for the players, staff and supporters who came up. The game developed exactly as we wanted.
"To be fair I had my head in my hands when Clifford hit the bar and when I turned round it was in the back of the net for our first goal.
"I was up at 4 am trying to work out systems and with plenty of adrenaline. We have taken a rollercoaster ride and got off safely."