Warriors fought back from a 13-0 deficit for a 21-16 win over the Bulls at a sold-out Loftus Versfeld.
Tries from Scott Cummings, George Turner and Huw Jones, all converted by George Horne, earned the title, adding to the Scotstoun club's 2015 success.
Having finished fourth in the regular season table, Warriors beat last year's champions Munster away from home seven days ago before delivering their first win on South African soil since 2018.
"It will be hard to beat this feeling," said Smith, the former Bulls player who joined Glasgow two years ago.
"To come here in front of 50,000 supporters and win the way we did, it's a special accolade to the men.
"We've grown leaps and bounds, but again, this sensation and what we've achieved tonight will be an important ingredient going forward.
"We had a saying during the season that we don't have to lose to learn and they've embraced that mentality and they're pushing their limits, the parameters of what is expected at this level.
"Obviously I'm really proud of the way they went about their business and the way they stuck to a plan. I honestly think this is just a start and hopefully we can keep on progressing."
Smith revealed that he had served up a round of whisky to his players early one morning in the early days of his tenure at Glasgow to give players "a wee taste" of future celebrations.
He explained: "Seven o'clock, I rolled out whisky for everybody. I said 'savour that, keep that in your mind. And one day, as soon as we can win something, we'll have an evening where we can have a dram of whisky and a cigar'. Tonight, I think these guys need it."
Glasgow were trailing by 13 points up until Cummings crashed over in first-half stoppage time, while the final ended with a yellow card for Tom Jordan and Bulls throwing everything they had at their depleted visitors.
However, Smith insists he always had faith that his side would prevail.
"I had calmness," he said. "I have belief in the boys and the process. We've been talking about process the whole year.
"We want to be the best version of ourselves. That's what we strive to be. We don't play opposition, we play against ourselves. If you look yourself in the mirror, you must be honest and understand that you've given a go once you retire.
"They must keep on giving a go and that will be our challenge now. How often can we repeat a performance like that?"