Mowbray told Sky Sports 3: "We played on the front foot. We pushed them back. I think we felt we had done enough to come away with three points."
He was frustrated, however, after conceding an equaliser less than three minutes after taking the lead.
Scott McDonald came off the bench to open the scoring with a 79th-minute header. Lee McCulloch headed in Steven Davis' corner two minutes later.
Mowbray said his side were aware of the danger posed by Rangers at set pieces and he was "disappointed" that the equaliser came from a corner.
Asked if Celtic could still catch Rangers, Mowbray said: "Undoubtedly. It's less than a task than they had last year."
Rangers boss Walter Smith accepted that his side was lucky to come away with a point.
He said: "We were very fortunate to get a draw. Celtic started the better side."
Smith said Rangers had "got away" with a point after not offering an attacking threat.
He added: "Celtic played very well. It was a tough afternoon for a lot of our boys.
"I felt the second half we did a little bit better. We are delighted to have got a point from a game we should not have. We were not with the pace of the game at all."