Charalampos Kostoulas, Maxim De Cuyper and Mats Wieffer put the tie beyond doubt with first-half finishes, before Olivier Boscagli added a deflected fourth late on to complete the rout. HĂźrzeler credited an early breakthrough for setting the tone.
"The early goal is always helpful, and I think we had a very professional approach today. Credit to the players because they pressed intensely and won a lot of personal duels," he said.
"They forced Tromsø into mistakes, and we kept another clean sheet. We maintained high standards in and out of possession and scored some nice goals.
"The approach we had was the key to the success. On top of that, early goals help, so overall, it was a very satisfying performance."
The result secures Brighton's place in Friday's draw for the Conference League league phase, offering the club regular European nights for the first time in this competition.
"I'm very happy for the whole club and everyone involved in that journey," HĂźrzeler said. "On top of that, I'm very happy for the players and the fans. They created another great atmosphere here tonight. It was very energetic and special, so thanks to the fans for that.
"We are very grateful for this opportunity to play in Europe. We know it will be a big challenge we have to face, but we seem to be ready. The key will be maintaining these standards and this intensity in the Premier League, but also in Europe. Hopefully, we are prepared for that. We will try to give our best, go game by game, and see what happens."
Jack Hinshelwood's absence from the squad was one of two changes from the win over Villa.
"He had a small issue in training yesterday, so we have to go day by day with him and see how it progresses over the next few days. I can't give you a clear schedule for when he will be back," HĂźrzeler said.
Matt O'Riley made his first start in a year in Hinshelwood's absence, and HĂźrzeler was pleased with what he saw.
"It wasn't the injuries that forced this decision. We still had other opportunities to replace Jack. It was a decision made because Matt deserved it through good training sessions and how he presents himself as a player, a person, and a team player," he said.
"He had a really good approach today. It isn't easy coming into the team for such a big, important game for the club, but he did his job well. He struggled at the end with the physicality, but he sacrificed a lot for the team, and I really want to emphasise that because it's what we need here and what we all appreciate.
"Let's see what we decide regarding his future, but overall, I'm happy to have him."