Valverde hails Bilbao's comeback
Ibai cancelled out Jese's opener at San Mames in a controversial La Liga match which saw Cristiano Ronaldo sent off.
Valverde, whose side are fourth but a long way behind the top three, said a draw was a good result given the opposition.
"After Cristiano was sent off, we went for the second goal, but it wasn't to be," he said.
"Although they were on top of the game at times, we've played how any manager would like their team to play.
"After such a good first half, we were playing thinking we could win, although Cristiano did have a chance to kill the game.
"We haven't lost, we keep adding points and the scale of our opponents also must be recognised."
Bilbao are six points clear of fifth-placed Villarreal but 11 behind Real, who sit third in a tight title race.
Valverde's men are unbeaten in 12 league matches at home and the coach said they needed to bring that form on their travels.
"We've stood up against the top teams. We're a difficult team to beat at home," he said.
"The fans can connect with the team and we're strong here because of that. It's the sort of form we want to transfer away from home too.
"The best thing is that the team compete in games like this, even knowing the potency of our opponents."
Valverde would have liked Bilbao to create more opportunities, saying they never produced their best going forward.
"We weren't able to get forward more. I thought we would but it wasn't because of caution," he said.
"Against Atletico we tried to play with two strikers and we didn't cause enough danger.
"Today my preference was to control our play going forward but we haven't done that as often as we would have liked."