Real move on to 60 points with 11 matches remaining while Barca have 63 ahead of their game at Almeria on Sunday.
Arjen Robben's powerful strike and a Gabriel Heinze header gave the champions a 2-0 lead after 34 minutes before Heinze put the ball into his own net and Fernando Llorente nodded past Iker Casillas to make it 2-2 at the break.
Bilbao's Fran Yeste was shown a straight red for shoving Casillas to the ground in the aftermath of Heinze's own goal.
Dutch international Huntelaar then squeezed a shot in at Gorka Iraizoz's near post early in the second half to make it 3-2 before racing on to Robben's pass and lifting the ball over the Bilbao keeper with just under half an hour left.
Real's victory, their 11th in 12 league matches, was a boost after Tuesday's 4-0 Champions League drubbing at Liverpool.
"The positive thing is that we came to a tricky ground and won and we are still in with a chance," Casillas said in a television interview.
"Let's hope Almeria will do us a favour tomorrow and we can keep this good form going."
Earlier, Valencia's hopes of a lucrative place in European competition next season were dealt a further blow when they were held to a 1-1 home draw by Recreativo Huelva.
Unai Emery's Valencia have not won a match since the beginning of February, slipping to eighth.
"As far as our reaction in the second half when we really picked things up that is how we want to continue," he told a news conference.
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