The Warriors claimed a 33-21 victory at Scotstoun, with captain Kyle Steyn producing a player-of-the-match display that included two tries.
Rowe felt Glasgow's desire ultimately proved decisive.
"We spoke all week that they weren't going to be a pushover team," Rowe told BBC Scotland.
"They came here and they tested us but I feel like we proved that we wanted it a bit more than them. I'm not saying that they didn't want it, but especially at home we just wanted it that little bit more.
"It wasn't perfect and we'll go again next week. [Steyn] was awesome tonight and the forwards put in a shift. Now we roll on to next week."
Glasgow will face either Bulls or Munster in next Saturday's semi-final at Murrayfield, and Rowe is hoping for a strong backing from supporters in Edinburgh.
He said: "We know the Warrior Nation will be there but we want as many people in Scotland, Scottish rugby fans, to come support us next week and make it a special occasion."