Spurs had to survive several scares - including William Gallas' goal-line clearance from Robinho in the first half - as Milan had the bulk of the possession in the second leg at White Hart Lane.
However, having won the first leg 1-0, the goalless draw at home proved sufficient to send Spurs into the quarter-finals and Dawson was delighted with the way his team-mates performed.
"Winning ugly? Definitely. The first half performance was back to walls, but we worked our socks off. Same in the second half. Look at it now - it's rocking," Dawson told ITV.
"Milan forced us to sit back. If you look at the players in our team, we're very attacking, but we showed a different side to us tonight. We'll enjoy tonight and see who we get in the quarter-final."
Peter Crouch, who scored the decisive goal at the San Siro, also stressed the defensive efforts throughout the side.
"It's huge. The club has never been in the Champions League before, then going to San Siro and winning, the lads defended magnificently tonight, great credit to them," he said. "I'm really pleased and looking forward to the next round. That's what it was all about.
"Sometimes in the Champions League you need to get results and that means defending from the front. We did that and I think over the two legs deserved to go through."