Smith was disappointed with the lack of penetration shown by his players against the Dons, who played with 10 men for the final 30 minutes after captain Mark Kerr had been shown a second yellow card.
Smith said: "Aberdeen started brighter but after about 20 minutes I thought we dictated the match, albeit with the help of Aberdeen being reduced to 10 men.
"Although we had a lot of territorial possession there wasn't any great creation in what we were doing.
"That has been with us at different times over the season and, from our own point of view, that was the disappointing factor more than anything else."
Smith refused to blame the defeat on his side's midweek exertions in their Champions League defeat to Stuttgart.
"We played on Tuesday night so I don't think that was a factor," he said.
"Aberdeen worked very hard and we have to give them credit for that, but the lack of creation has been in our game for quite a period of time and it was apparent again today."
Delighted Aberdeen boss Mark McGhee, who had to make a late change by bringing in Davide Grassi for Jerel Ifil who was injured in the warm up, believed his side deserved the win.
"I think we earned it when we had 11 men and went ahead as I thought we were the better team at that stage and the more creative," he said.
"They were always a danger, particularly with Kris Boyd who I'm a big fan of.
"You have got to be vigilant and we were. The boys showed a heroic commitment to keep them at bay after the sending off so all in all I felt we deserved the three points.
"But the most important thing for me was the first hour where we took Rangers on on their terms and I thought we were the better team."
McGhee singled out Derek Young, making his first start since a 5-1 home defeat to Sigma Olomouc in a Europa League qualifier in July, and Grassi, playing for the first time since fracturing a cheekbone, for special praise.
He added: "We had to change it about after Andrew Considine was injured in training yesterday and Jerel Ifil went down in the warm up.
"It was a big ask for them to come in today but they played very well. Davide has come back from a really serious injury and not played a game yet.
"I haven't really given Derek a chance since I came here but he has trained well and his attitude has been first class.
"I was pleased for them but it was a team effort in the end."