The visitors struck straight from kickoff with Hakan Calhanoglu feeding Sebastian Boenisch who sent the 24-year-old winger through to score.
"I am really happy for him (Bellarabi)," Leverkusen sports director Rudi Voeller told reporters. "This record is great for him. We did absolutely everything right in that first half and so did he."
Bellarabi returned to Leverkusen this season after a loan spell at Eintracht Braunschweig.
The previous record was 11 seconds, held by Ulf Kirsten, Giovane Elber and Paul Freier.
"It's great that the Bundesliga record will now stay at Leverkusen. That's great for us and for the player," Leverkusen chief executive Michael Schade said, referring to Kirsten's goal against Kaiserslautern in 2002.
Stefan Kiessling scored another goal from a Bellarabi pass with the last kick of the game for Leverkusen's season opening win.
(Reporting by Karolos Grohmann, editing by Tony Jimenez and Josh Reich)
