The Portuguese forward flicked Luka Modric's low drive into the back of the net in the 87th minute with the deftest of touches, to cancel out the first-half chip from Bilbao's Inaki Williams.
Real were set for defeat until Ronaldo, who also hit the bar with a header early on, struck with his 24th league goal of the season to snatch the draw that left them third on 68 points after 33 games.
"I'm not happy with the result, it's not positive for us. We played well, above all in the first half," Real coach Zinedine Zidane told reporters.
They are three points behind their cross-city rivals, second-placed Atletico Madrid, who play their game in hand at Real Sociedad on Thursday. Barcelona lead the way on 83 points, 12 clear of Atletico.
Raul Garcia wasted an excellent chance to double Bilbao's lead before Ronaldo's equaliser, striking the crossbar with Keylor Navas out of position after a scramble in the box.
"We did huge work. We knew that to win here the goalkeeper has to be great, they have to mess up, we have to have a perfect game," Athletic coach Jose Angel Ziganda said.
"We defended well but they close you in. It's a pity that so close to the end they equalised. I leave with the feeling that we have an honourable team."
(Writing by Rik Sharma,; Editing by Ian Chadband)
