Arsenal won the tie 2-1 on aggregate, with Bukayo Saka's first-half goal on Tuesday proving the decisive contribution across 180 minutes of intense, tight football. Atletico Madrid had chances across both matches but could not convert when it mattered.
Koke faced the cameras in a visibly emotional state immediately after the final whistle, yet his message was one of pride rather than bitterness.
"Very hurt, very gutted — but proud of my team and my people. We gave absolutely everything. We had chances to score and the ball just wouldn't go in. The same thing happened in the first leg. Football is about clinical finishing, and they were more clinical than us."
He refused to point the finger at referee Daniel Siebert, who failed to award Atlético two second-half penalty appeals — for a challenge on Giuliano Simeone and for an apparent foul on Antoine Griezmann — despite sustained protest from the visiting bench.
"I'm not going to talk about the referee. He tried to do his best. He'll know how he refereed."
On where Atlético fell short across the two legs, Koke was honest.
"Nothing to reproach anyone for. The team went for it and it wasn't to be. We had them on the ropes at various moments but they found the goal and they were decisive."
His broadest reflection was reserved for the season as a whole and for where this Atlético squad is heading.
"In the Copa and in the Champions we were very exciting, but in La Liga we weren't regular and we dropped points in games where we shouldn't have. This group is growing a lot and I'm sure that in the very near future they'll be fighting for everything."
Asked about his own future at Atlético, Koke, who turns 35 in January and whose contract situation is unresolved, kept his answer short.
"We'll talk when the time comes."
Tuesday's defeat likely brought down the curtain on Antoine Griezmann's Champions League career with Atlético. The 35-year-old, who is joining MLS club Orlando City in the summer, came off the bench during the match and was given a prolonged ovation by the travelling supporters.