Introduced in the 67th minute, the Uruguayan midfielder settled a remarkable Champions League contest with a superb strike in the 92nd minute, having watched his side come from behind twice.
Speaking after the match, Valverde praised the collective response from a team that refused to accept defeat.
"Congratulations to our team for coming back twice. It means we have good energy and a positive spirit," he said.
The midfielder explained that Madrid had anticipated AEK's defensive approach, with the hosts protecting their penalty area and allowing opportunities for shots from outside the box.
"We knew they defended very well inside their area and left space outside the box for us to shoot. Today the shots went in, which doesn't always happen."
Valverde had already highlighted the difficulty of the contest in his interview with Movistar, describing a match in which Madrid enjoyed possession but struggled to create clear opportunities.
"It was the kind of game we expected, very scrappy and difficult. I think we dominated, but we struggled to get into dangerous positions. Congratulations to the team for the effort and the work, and we have to keep going like this."
He admitted that conceding the opening goal had affected Madrid, but was pleased with the way the players maintained their approach.
"The goal affected us, but the team continued playing its game. Sometimes it is difficult when you concede, but we put in a lot of effort and I'm happy for the whole group."
The captain also credited the coaching staff and the squad's collective belief for the late turnaround.
"It is credit to the coaching staff and to us for trusting all our team-mates. You can see that in the final minutes because the team never stops trying. It looked as though it would be very difficult after we conceded the second goal, but we came back again."
Valverde was reluctant to take individual credit for his winning strike, instead thanking his team-mates for giving him the confidence to shoot.
"Sometimes I think about it too much. I thank my team-mates because they give me that encouragement to hit the ball."
He also defended Vinícius Júnior, who was substituted late in the game, insisting the Brazilian should continue taking risks even when things do not go his way.
"The important thing is that, even when he makes mistakes, he keeps trying and keeps contributing. He gives the team that energy, he has a good connection with everyone and the good things will come."
With Madrid continuing to collect victories, Valverde said the atmosphere around the squad was positive.
"You work with less pressure and enjoy things because they are going well. And tomorrow, it's back to training."