Victory for Madrid, who have smashed the La Liga scoring record, means coach Jose Mourinho has now won the title in four major European nations, a feat that might never be repeated.
It went some way to easing their pain after a defeat on penalties to Bayern Munich in the Champions League semi-finals last week.
Lionel Messi's record-breaking hat-trick in Barcelona's 4-1 victory at home to Malaga in the earlier kick-off had kept alive Catalan hopes of catching their bitter rivals but Real swept Europa League finalists Bilbao aside.
They have a seven-point lead with only two games left.
Mourinho becomes the first man to win league titles in his native Portugal (Porto), England (Chelsea), Italy (Inter Milan) and Spain and has ended Barca's three-year stranglehold on the title, the club where he was an assistant coach in the 1990s.
"It has been a very long and hard season but we have taken a step forward in terms of quality compared to last year," Real captain Iker Casillas said.
"It was tough to end Barca's run as this is a new project and we are a young squad but I think we already improved last year and even more so this year," added the Spain goalkeeper, as his team-mates celebrated wildly on the pitch nearby.
Real have only lost twice all season, 1-0 at Levante and 3-1 at home to Barca, and have amassed a record 115 goals with 30 conceded.
Their success has been largely down to the awesome firepower of their front line, with Ronaldo racking up 44 goals, Higuain netting 22 and Karim Benzema 20, the first time three players from the same club have passed the 20 mark.
Mourinho also gave an expensively-assembled group of players the self-belief that allowed them to get the better of Barca, widely regarded as one of the best teams of all time.
Barca's home form let them down and they dropped 16 points on their travels to Real's seven before a 2-1 defeat by Real in the 'Clasico' at the Nou Camp last month effectively ended their title bid.
"We are very happy after winning a very closely-fought championship against a very strong Barcelona," defender Sergio Ramos said.
"Madrid deserve the title and we dedicate it to all the fans."
The match at the San Mames got off to a blistering start, with both sides pouring forward in search of a goal.
Real should have been ahead in the 12th minute when the referee ruled Javi Martinez had handled the ball in the area and awarded a penalty.
Ronaldo, who missed a spot kick in the Champions League semi-final shootout against victorious Bayern Munich last week, attempted a dinked shot down the middle of the goal and goalkeeper Gorka Iraizoz diverted the ball over the bar.
It was the first time the Portuguese had fluffed a penalty in La Liga in 12 attempts since joining for a record fee from Manchester United in 2009 and drew howls of derision from the Bilbao faithful.
Real were ahead four minutes later, however, when Ozil picked out Higuain on the edge of the area and the Argentina forward smashed a shot into the top corner.
They doubled their lead in the 20th minute when Ronaldo made amends for his penalty miss with a superb pass into the path of Ozil and the German international sidefooted past Iraizoz.
Their third came five minutes into the second half when Ronaldo was left unmarked at a corner and he nodded home. It was the Portuguese's 44th league goal of the season leaving him two behind record La Liga top scorer Messi.
Bilbao, who play La Liga rivals Atletico Madrid in the Europa League final on May 9, were twice denied by the crossbar in the second half before Martinez was booked for a second handball and sent off.
As the Real players flung Mourinho in the air on the San Mames pitch, thousands of Real fans already gathered at the Cibeles fountain in the capital began celebrating the end of four years of disappointment.