Horne broke DTH van der Merwe's record with Glasgow's bonus-point try in the 42-0 win played in 50mph winds at Scotstoun.
"I'm really proud of him," Smith told Scotland Rugby News.
"The energy that he brings to the team, what you see out there is exactly how he's every day at training. He's positive and works really hard.
"He's a very good rugby player as it is, but he's learned how to use the system to his benefit, the way we play. He's been converting some very good line breaks. He's a real asset to the club."
After scoring six first half tries, Warriors threatened but didn't add a point in the second period, but Smith was pleased with the full 80-minute performance.
"I'm pleased with the whole game," he said.
"I don't think people realise how tough it is to play into this wind. We couldn't kick the ball once. Stafford kicked it once in the second half. To keep the ball on your side of the pitch, you end up having them having 14 in the front line. You can't find space.
"There was good application of what we wanted to do.
"In the first half, obviously, we set the tone in what we wanted to achieve. We're really proud of the effort tonight."
Smith praised his team's defensive performance against a usually dangerous Lions side.
"Again, the zero says a lot," said Smith.
"They do score tries, they score a lot of tries. They ripped us apart last year - in five minutes, they scored four tries last season. We were aware of that. They are a dangerous team."
Jack Mann dropped out of the squad late on with a calf injury, and Smith said there will be further assessment before it's known if the big forward to will be available to face Leicester next weekend.
"He hurt his calf on Tuesday. It just stiffened up. On Thursday, he actually felt good, but for where we are in the season, and with Jack Dempsey out, we just didn't want to take a risk. We might need him going forward. For now, it was more precautionary."