Valverde bemused by Iraizoz's disallowed goal
Malaga led 1-0 heading into second-half stoppage time at home but Valverde's visitors looked to have secured a point when Bilbao goalkeeper Gorka Iraizoz powered a header into the top corner of the net from a free-kick in the penultimate minute.
But the referee disallowed the goal with Bilbao defender Aymeric Laporte seemingly offside, although he was barely involved in the play, while Valverde claimed there may have been a foul, even though he did not see anything.
The 50-year-old coach remained diplomatic after the match but could not hide his disappointment and confusion.
"Everyone sees things from a different perspective, although it's difficult for a player like Iraizoz to score a goal like that and have it annulled," Valverde said.
"I don't want to talk to much about the referees, he made his decisions, some were wrong and some were right.
"He must have seen a foul, on someone for something, that's the feeling we have."
Malaga took the lead 10 minutes before half-time when on-loan Liverpool attacker Luis Alberto saw his penalty saved by Iraizoz but managed to tuck the rebound into the net.
The home side then had to survive a nervous finale as they lost two players - Duda and Vitorino Antunes - to late red cards, before Iraizoz's towering header was ignored.
Despite the loss, Valverde insisted his side controlled their opening game of the 2014-15 La Liga season.
"Malaga were waiting to surprise us in the counter attack, with very good players," he said.
"The game was under control, but unfortunately, they drew a penalty. We dominated but we lacked sharpness in the first half and in the second half as well and in the end there were expulsions, a disallowed goal and there's not much else to say."