Nico Williams and Robert Navarro struck within two minutes of the restart before Oihan Sancet completed the rout late on, as Athletic Club overwhelmed their visitors after a goalless first half in which Pablo Barrios had hit the post.
"I'm very happy and satisfied. It was a very intense game with great moments and it's a great result," Terzic said.
"I liked everything about my team and the atmosphere at San Mamés was incredible. It was incredible how we approached it, and after the hydration break and half-time the team was incredible."
Terzic also gave his squad an extra day off.
"It's always good to win a game, and to do it at home too. I thank San Mamés for the atmosphere," he said. "Tonight we'll sleep very satisfied, but these six points are only the start of the season. Now we need to prepare for the next game."
Terzic singled out several performances from the win.
"We can talk about Nico, but also about Oihan, who played in four different positions. He can be one of the best players in the league. And I can also talk about a good save from Unai," he said. "I'm very happy with the whole squad and also with the work of the entire technical staff."
Asked about the varying structures shown this season, Terzic insisted the philosophy remained collective.
"It's not my Athletic, it's our Athletic, and what the people want to see. We want a team ready to fight and win matches," he said. "Different structures are something necessary in the world of football."
Simeone, whose Atletico Madrid side suffered their first defeat of the campaign, pointed to a costly lapse either side of half-time.
"The game was what you saw. A good first half, and two minutes of poor attention and decision-making that we should have resolved better. Athletic took advantage of that moment of disconnection. They won the game fairly," he said.
The Argentine reflected on how quickly the contest turned.
"The game was going very well heading into the second half. But we didn't resolve those two minutes well — though responsibility lies with the coaching staff for not putting the players in a position to resolve it," Simeone said. "We are building this team. We're going to go through complex situations. We started well, but those two minutes today gave us a blow to react to."
Despite the scoreline, Simeone was measured about Athletic's threat.
"We know their coach and he has things clear. Apart from Julián's chance, we didn't have any clear opportunities. Athletic are going to fight for the top positions," he said.