Dortmund ended the Group F contest with 10 men after goalkeeper Roman Weidenfelder was sent off and coach Juergen Klopp was sent to the stands after an angry outburst.
There was better luck for Dortmund’s Bundesliga rivals Schalke 04 as they scored three late goals to overcome a rugged Steaua Bucharest side who dug deep for 67 minutes.
Stunning strikes by Theo Walcott and Aaron Ramsey gave Arsenal a 2-1 victory at Olympique de Marseille to continue their record of never having lost a European game away to a French club.
Arsenal have won 10 successive away matches in all competitions and manager Arsene Wenger said they were full of belief for what is a difficult group including Napoli and Dortmund.
“I knew it was important to get three points here if we want to have a chance,” Wenger said. “I said yesterday we needed 10 points at least so to start with three is good.”