Bellingham equalised two minutes into the second half after Sasa Kalajdzic had broken the deadlock.
Marco Reus’ first league goal since December put Dortmund ahead for the first time in the match.
Daniel Didavi levelled for Stuttgart, but teenager Ansgar Knauff restored the visitors’ lead with his first Bundesliga strike.
Edin Terzic’s side remain seven points behind fourth-placed Eintracht Frankfurt, who beat fellow European hopefuls Wolfsburg 4-3 earlier on Saturday.
Dortmund trail Manchester City 2-1 going into Wednesday’s Champions League quarter-final second leg at Signal Iduna Park.