Swinson delighted after historic triumph
The Scotland second row forward scored one of six tries as the Warriors inflicted a record European defeat on Leicester Tigers at Welford Road. Gregor Townsend's side have advanced from Pool One as one of the best runners-up after the 43-0 triumph.
And Swinson was delighted to make history for Glasgow.
"I feel fantastic, it was our goal to get out of the pool at the start of the season and we've managed to achieve that which is a great feeling. Now we go onto the next stage and give that a go.
"We focused on ourselves a lot this week. When we play our rugby, we can beat anyone and we saw that today. It was a really good team performance. The forward pack worked extremely hard and the backs were great as a unit and that's what put them to the sword," he said.
Glasgow scored a total of 85 points against Leicester in their two pool clashes. Scotland flyhalf Finn Russell was central to their last-eight qualification and he hailed his forward pack for laying the foundations for a destructive attacking performance.
"We knew how physical the game was going to be today and credit to the boys up front because they made it easy for us. The set-piece and lineout drive was outstanding today.
"We looked at ourselves a lot this week and we knew how tough it would be to come down here and get a win. We knew we would have to play with tempo and play quick, attacking rugby and we did that for the full 80 minutes," Russell said.