The Leverkusen attacker struck after just nine seconds and then set up Stefan Kießling in injury time as the visitors won in Dortmund for the second season in a row.
Bellarabi's strike was two seconds quicker than the previous record held by Bayern Munich's Giovane Elber and Leverkusen's Ulf Kirsten, both of whom scored in the 11th second of a Bundesliga game.
"My team did a lot right," Leverkusen manager Roger Schmidt said. "We attacked our opponents early. We played a great game and at the end of the day we deserved to win."
The visitors needed just three passes after the kick-off to find the net. Sebastian Boenisch picked out Bellarabi on the edge of the penalty area and he turned inside Matthias Ginter before firing beyond Mitchell Langerak.
Dortmund were shellshocked and Leverkusen gave them little time to find their rhythm.
New recruit Hakan Calhanoglu drove a free-kick wide for Leverkusen before Boenisch screwed a shot well off target.
Last season's runners-up improved after the interval but Leno brilliantly saved a header from Ginter with six minutes remaining and denied Ciro Immobile a debut goal in the final minute of normal time.
Kießling then sealed the win five minutes into stoppage time after Bellarabi won the ball from from Dortmund left-back Erik Durm.
"For the first minute, I can't give any explanation," Dortmund manager Jurgen Klopp said.
"On account of that first minute, the game became a bit strange."