Yuri Berchiche was sent off before the quarter-hour mark, and Óscar Sosa Oso punished the resulting disadvantage soon after, before Chidera Ejuke and Isaac Romero struck early in the second half to put Sevilla out of sight.
Gorka Paredes' late reply proved only a consolation as Luis García Plaza's side made it six points from six to open the LaLiga season.
Terzic, in his first competitive match in charge of Athletic, was reluctant to dwell on the sending-off.
"I'm not going to talk about the referee's decisions, only about our performance," the German told reporters. "We made the mistake of being too optimistic in a passage of play that ended with Yuri's red card.
"Then we conceded three goals because we made mistakes, and it's not easy to win that way. But we never gave up. Nobody accepted the result."
The coach praised Álex Berenguer's improvised role at right-back following the reshuffle forced by the red card.
"He's done an excellent job in that position. It was our first idea because going forward he could also go one-on-one. At half-time we tried to adjust things to try to have more players in the box and cross the ball."
Terzic pointed to Aingeru Gere and Iñaki Canales for their work covering ground while Athletic Club played a man short, but was clear that his side's attacking play needs work.
"We have to improve offensively so that everyone knows where they have to be and how they have to move and spread out. We haven't been efficient there. We have to increase the moments where we have a lot of players available to score, not just one player in a position to do it."
Despite the result, Terzic took heart from his side's response after going 3-0 down.
"I saw a good reaction after two hard blows, the red card and the first goal. Seeing that the team didn't give up is important to me as a coach. Obviously we have to look for ways to improve and win, but I can't ask for more in terms of the energy the players gave."
García Plaza, whose Sevilla have started the campaign with back-to-back wins, admitted his side grew nervous once Athletic pulled a goal back.
"It's very difficult to win two games in a row. We were very good until 0-3, but when they scored we got very nervous and suffered more than we should have. They're very good, because individually they have players of a very high level, world champions, and the team has to keep working on those things — when you win the ball back, having more composure, finding the free man."
The Sevilla boss praised his academy graduates for stepping up, with Isaac Romero setting up Ejuke's goal shortly after coming on at half-time.
"The academy has to take a step forward, and I'm very happy about that. Isaac, for example, has recovered well, came on and had a spectacular second half."
Looking ahead to hosting Atlético Madrid next, García Plaza added: "We have six points already and nobody can take that away from us. Let's work for the game against Atlético — there'll be a fantastic atmosphere at our stadium, and we'll try to make it three."