"After a disappointing home result on Wednesday (defeat to Hibernian), it was important that we won today's game," Smith told BBC Radio Scotland.
"We did so and I thought, in the first half especially, we played very well."
Kenny Miller headed Rangers in front and missed a penalty after Vladimir Weiss had scored the home side's second.
"We're just delighted that we've managed to win what was always going to be a tough game," said Smith. "No team loses in the manner Aberdeen did last weekend at Celtic Park and then come to Ibrox and set their stall out in the same manner.
"Today, they stuck to their task throughout the 90 minutes.
"If the third one goes in, there's obviously a little bit of pressure on them to change a little bit, which is what we would've hoped to happen but we missed it. We kept going, we kept playing. We had one or two chances in the second half that could have added to the score.
"I'm just delighted that we managed to get the breakthrough early, it was a terrific goal by Kenny Miller, a good header.
“When teams come and set themselves up the way Aberdeen did today, it's sometimes an awkward situation when you're trying to commit a lot of players forward to get a goal.
"We play in the same manner in the European games so we know what to expect when we come up against it.