The German head coach, speaking to TNT Sports, said his side's first-half performance set the tone for the result.
"It was very intense and tough. Brilliant first half in everything. The way we won balls and the possession high up," Rose said. "In the second half, we needed to suffer because we were not so sharp. We need to talk about the second half."
Rose pointed to Bournemouth's pressing intensity as the foundation of the win, and suggested his side had missed the chance to extend their lead further before the break.
"That's Bournemouth football. We want to be active and defend forward. We want to create chances," he said.
"We had a header to be 3-0 up. The game is 90 minutes and the first half was really brilliant."
The second-half dip, Rose explained, was less about tactics than about maintaining the mentality his team requires to play the way he wants.
"It's not about the games because we didn't play so much so far, we just started. It's about the mindset," he said. "We lack a bit of intensity and if you lose this kind of thing in our style, then we know. If you get tired, then we need to change with substitutes."
Despite the second-half wobble, Rose was delighted with the overall outcome on a landmark night for the club.
"It's a brilliant win in our first European game," he said. "It's brilliant for the club and the 2,000 fans from Bournemouth."
The result marks Bournemouth's first competitive European fixture in the club's history, following their sixth-place finish in the Premier League last season under Rose's predecessor, Andoni Iraola — a campaign that also included an 18-match unbeaten run to close out the season. Rose was appointed in April to build on that progress, arriving with previous European experience from his time in charge of Red Bull Salzburg, Borussia Mönchengladbach and Borussia Dortmund.
Thursday's trip to the Basque Country was the first of eight league-phase fixtures for Bournemouth this season, a group that also includes home games against Milan, Viktoria Plzeň, Sturm Graz and Hapoel Be'er Sheva, along with further away trips to Celta Vigo, Sparta Prague and Lillestrøm. Rose's side will look to build on Thursday's result when the league phase resumes next month, with the manager having stressed all season that adapting to the intensity of continental football would be part of the learning curve for a squad experiencing European competition for the first time.