With Lionel Messi oddly quiet, Neymar opened in the 19th minute from close range after receiving Andres Iniesta's pass as the Brazil striker outshone fellow newcomer and adversary Gareth Bale.
"Scoring a goal in a clasico is very special," Neymar said. "It's a game that everyone wants to play. It was like a dream come true."
While Neymar has lived up to his 57-million-euro transfer this summer, questions over the wisdom of Madrid's 100 million euros payout for Bale will continue after he failed to produce a shot on goal before being replaced by Karim Benzema with half an hour to go.
Benzema went on to hit the woodwork as Madrid threatened to equalize before Alexis took Neymar's pass and sealed the victory in the 78th with a brilliant chipped shot.
Substitute Jese Rodriguez got Madrid's consolation goal in injury time after a lightning counterattack.
Neymar followed in the footsteps of former Brazilian standouts Romario and Rivaldo, who scored in their first matches for the Catalan club against Madrid. And his playmaking echoed Ronaldinho's assist in his first clasico.
Barcelona coach Gerardo Martino, also making his clasico debut, was careful not to overpraise Neymar, while also giving credit to Messi for his work in defence.
"The transcendence of this game and the goal he scored magnified the level of his performance," Martino said about Neymar. "But I think he has had better games than tonight. As far as what (Messi) did for the team to recover the ball, I give him a superlative mark."
Also, Celta emphatically broke a four-game losing streak with a 5-0 rout at Malaga. Levante rolled to a 3-0 win at home over Espanyol, and Granada won 1-0 at Elche.
Madrid had improved in its recent visits to Camp Nou, including a 3-1 win in the Copa del Rey last February, but Barcelona restored its home dominance over its traditional rival for the league crown.
The victory was Barcelona's ninth to go with one lone draw in the league this season, lifting it a provisional four points clear of Atletico Madrid which hosts Real Betis on Sunday.
Madrid was left a full six points adrift of the pacesetters and with reawakened doubts over the direction of the team under new coach Carlo Ancelotti after losses to both of its league challengers. It lost 1-0 to Atletico last month.
"I think we showed good football in front of a fantastic team," Ancelotti said after his first clasico. "At this moment it was a good test. We are improving."
Bale made his second start of the season since being hampered by muscle injuries, but he didn't prove to be effective in attack with Cristiano Ronaldo.
Ancelotti further tweaked his starting 11 with Sergio Ramos in midfield, while Martino chose Cesc Fabregas to accompany Messi and Neymar in attack.
Neymar struck the ball into the arms of Madrid goalkeeper Diego Lopez in the game's first shot on goal in the 17th.
Two minutes later, Iniesta dribbled into the area and laid off for Neymar on the left flank. Madrid appeared to have him hemmed in, but he used his right foot to slip a shot between the legs of Dani Carvajal, with the ball taking a deflection off the defender's thigh before trickling in.
It was Neymar's fourth goal in 14 matches since joining from Santos this off-season.
While Neymar sparkled, Bale was limited to a pair of harmless long-range attempts as Madrid failed to muster a shot on target until a minute before halftime.
Madrid gained ground as the second half wore on and Barcelona appeared to tire, leaving Benzema unmarked to rip a shot off the woodwork in the 72nd.
But Neymar helped Alexis secure the three points when he spotted the Chilean's run after he came on for Fabregas in the 70th.
Alexis pulled up outside the area with Raphael Varane in front of him and coolly lofted the ball over the defender and the diving Lopez, who was caught just off his line.
"I didn't have any other option," Alexis said. "I saw the goalkeeper coming forward and I had to try it. Fortunately it went in."
Neymar received an ovation when he was substituted six minutes from time before Rodriguez fired under Victor Valdes, who plans to leave Barcelona next summer and could have played his last home game against Madrid unless the teams meet in the Champions League or Spanish cup.
Before the match, Camp Nou displayed a giant mosaic with the message "Forca Tito" ("Stay Strong Tito" in Catalan) in support of former coach Tito Vilanova, who had to step down this summer as he fights throat cancer.