Barca went behind on the half-hour mark following Tamudo's opener, but responded just before half-time thanks to a controversial Messi effort.
The Argentinian struck again after the break, but Barcelona were denied in the dying minutes as Tamudo equalised and Real Madrid also came from behind to draw at Real Zaragoza and stay on top of the Primera Liga.
"It's a very big blow," admitted Barca coach Frank Rijkaard afterwards. "It feels like a defeat.
"It is a missed opportunity and that's difficult to accept. But we must keep fighting and show our pride.
"We could've have killed the game off. I never thought the title was in our hands, but I preferred to attack less and keep what we had."