Braces from Lionel Messi and Thierry Henry, coupled with goals from Carlos Puyol and Gerard Pique crushed the reigning champion, which is now seven points adrift of the lead with just four games left.
If Real fails to beat Valencia next weekend and Barcelona defeats Villarreal at Camp Nou it will be crowned league champion.
"It is not easy to win at the Bernabeu and winning like this against a team of Real´s caliber is incredible. It is a memorable night," said Guardiola.
"It is one of the happiest days of my footballing life. I am so happy because we have made a lot of people happy. We have taken a giant step toward the title and are so close now.
"There was only one way to approach this game and that was to be brave and try to win the match by attacking. We went behind and then had to go forward and create some chances which we did.
"I have to give credit to Real because they never give up and only a team of their stature could prevent us from winning this league."
Real was on an unbeaten 18-game league run stretching back to mid-December when Barca beat it 2-0 at Camp Nou, but was outclassed in a one-sided Clasico that will be remembered for many years.
Gonzalo Higuain put Real ahead after 14 minutes but Barca fought back to silence the home crowd with a dazzling display.
Barca had to do the guard of honor at the Bernabeu last season to salute Real as champion and then lost 4-1, so this was sweet revenge.
"After what happened last season we had to do something," said Henry. "Last season we played here and it was a difficult time. This year it was better. We wanted to win like we did and have managed to do it.
"It is like a party but there are games left and Madrid has shown that it doesn´t want to surrender the title."
Barca created a glut of chances and could have got into double figures had Real goalkeeper Iker Casillas not been in fine form, thwarting Messi on three occasions alone.
Casillas was still helpless to prevent the six he conceded and it is the first time Barca has scored six at the Bernabeu in history.
Barca has scored an incredible 100 goals this season at an average of almost three goals a game and is now targeting Real´s all-time goals record of 107 set back in 1990.
But first there is the matter of trying to reach the Champions League final with Barca facing Chelsea in the semifinal second leg at Stamford Bridge on Wednesday.
The first leg finished 0-0 and Guardiola urged his side to be brave in London.
"This win gives us a lot of confidence for the Chelsea game," insisted Guardiola. "Rome (where the final will be played) is so close. We don´t feel tired and just have to be brave and go for it."
It is an incredible first full season in management for Guardiola, promoted from the youth team in the summer, as he tries to guide Barca to an historic league, Kings Cup and Champions League treble that no Spanish side has ever achieved.
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