Inter won 2-1 on Tuesday to wrap up the round of 16 tie 4-1 on aggregate.
Dumfries said: “I’m proud, it’s a special moment for me. First time as the captain at Inter, I’m really happy. It was a special evening, but I hope (Stefan) De Vrij will recover quickly, he’s a great player."
De Vrij broke down in the warm-up, leaving the captaincy to his fellow Dutchman.
“(It was) a great game, a great evening for us. I’m happy to go to the quarter-finals against Bayern who are a great team.
“I feel good, I want to carry on like this. I’m happy because I’m playing a lot and playing well.
“We’re a strong team, everyone wants to beat us. We play with a lot of concentration, and we have to keep that up.”