The Portuguese forward flicked Luka Modric's low drive into the back of the net in the 87th minute with a deft touch, his 24th league goal of the season cancelling out a first-half chipped goal from Inaki Williams for the visitors.
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.
They are three points behind cross-city rivals Atletico, who play their game in hand at Real Sociedad on Thursday. Barcelona lead the way on 83 points, 12 clear of Atletico.
"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 said.
Raul Garcia wasted an excellent chance to double Bilbao's lead before Ronaldo's goal, striking the crossbar with Keylor Navas out of position after a scramble in the box.
Bilbao came close to earning their first victory at the Bernabeu since 2005 but Ronaldo denied them, finally netting for Real after the hosts wasted a string of chances.
Athletic goalkeeper Kepa Arrizabalaga made superb saves to deny Raphael Varane and Marco Asensio, but stood no chance with Ronaldo's equaliser.
While Real were on top during the game, it seemed hearts and minds may have been drifting towards next Wednesday's Champions League semi-final first leg against Bayern Munich in Germany.
Zidane picked a strong side, with his team not due to play their league match against Sevilla at the weekend because the Andalusians are playing in the King's Cup final against Barcelona.
"It's true that we are thinking about Bayern Munich, but not when we were playing. We have a week to work on that, we're doing well. We're confident and we are playing well," Real winger Lucas Vazquez said.
Earlier on Wednesday, fourth-placed Valencia lost 2-1 at home to Getafe and had Dani Parejo sent off for an elbow on Jorge Molina.
The defeat left Valencia three points behind Real Madrid.
(Writing by Rik Sharma; Editing by Ian Chadband)
