The Championship clash at St Andrew's finished as a goalless draw, but both sides created enough chances to win the game.
Birmingham were arguably the better side for the first hour of the encounter, with Jota guilty of wasting a gilt-edged opportunity after the break.
"We could have [won it]," Cotterill told Sky Sports News. "I know they've hit the bar with a couple of chances but they were on the counter-attack, I thought we were the better side in the second half but the stoppage in the game slowed us down a bit.
"From a personal point of view it's always important when you're new into a club to not get beaten by your local rivals at home. You can see that Villa have improved this year, Steve has had time there and now they're moulding into a team. They've got some very good players but I thought we deserved to win it.
"We caused them problems today and hopefully we can carry on doing that to other teams. I've said to the lads that it's all well and good getting yourself up for Cardiff at home when they're top of the table and for a local derby, but we had that blip in the middle there at Millwall. The Brentford and Barnsley games are very important now before the international break."
Birmingham remain 21st in the Championship table following the stalemate.