Glasgow produced an all-action first-half display, crossing for four tries to surge into a commanding 26-7 lead at the break. Despite being reduced to 13 men at one stage, with two players serving time in the sin bin as Clermont mounted a fierce comeback, the Warriors kept their composure to seal a 33-21 win and book their place in the Round of 16.
Looking ahead, Glasgow travel to face Saracens next Sunday, with the opportunity to secure top spot in Pool 1 and earn a home tie in the knockout stages.
"Obviously very proud of the boys, they worked really hard for it," Smith told Premier Sports after the match.
"I always knew Clermont was going to come out in the second part of the game. They're a proud team and in front of this crowd it's always difficult.
"They did exactly what we thought. I thought we handled the yellow cards well, which allowed us to compete until the end.
"I think the physical application (pleased me most).
"The forwards obviously didn't get much reward from the maul. The defence close to the line was exceptional. But in general I must say a very good performance among us.
"Saracens are obviously a tough team, we know that.
"In preparation for this game we already knew what lies ahead next week. It's a Sunday game, we will make full use of this week's recovery to make sure we roll our product out.
"The important thing is to keep the expectation at bay."