Smith told BBC Scotland: "All the people make it special here. We are really proud. We played for them.
"It was part of our pre-match conversation. We want to make Scotstoun proud, and we heard them tonight. So a really, really good evening.
"[Saracens] are a quality team, and they've got some excellent game-drivers with a lot of experience there. They've got good a pack of forwards.
"Maro Itoje disrupted our set-piece by himself at some stages. So, a tough team. They did well to stay in the fight. You can see that they're a proud team, obviously, and a lot of character.
"It was good for us to show that we can play an all-round game, both sides of the ball.
"The defence in the second part of the game was fantastic. We worked really hard to apply ourselves physically, and to be fit enough and resilient enough to stay in the fight. It was a pleasure to watch.
"[George Horne] knows I'm really proud of him. Everybody here is really proud to have him with us.
"The way he goes about his business and how much he believes in the club is obviously seen by everybody out there. I'm really, really proud of him and I hope there's another 100 to come.
"We've had tough teams. Nobody would have bet on us before the start of this campaign that we would end up where we did.
"I'm happy with that, first of all. But we also know this is just the start. There's more than half the season still left and it's going to be important to keep everybody healthy and to keep the energy levels high so that we can continue to push hard in this competition."
Franco Smith's post-match media conference