The 25-year-old's clipped shot squirmed in, thanks to some flimsy goalkeeping from Olympique goalkeeper Steve Mandanda, with just three minutes to spare to send last season's finalists through.
Dortmund missed a hatful of chances against the negative Frenchmen, who finished Group F with 0 points from their six games.
Robert Lewandowski's fourth-minute opener was soon cancelled out by Souleymane Diawara, before Dimitri Payet collected two yellow cards within two minutes and was sent off in the first-half.
Dortmund laid siege on Mandanda's goal in the second 45. Jakub Blaszczykowski's header was well saved, Marco Reus struck the post, Lewandowski missed an open goal and Reus, again, somehow skied the ball with the goal at his mercy.
A draw would have been enough with Napoli and Arsenal firing blanks, only Gonzalo Higuaín struck on 73 minutes in Naples and Dortmund were heading out.
Desperate, Klopp introduced Julian Schieber and reverted to 4-4-2 in a bid to claim the elusive goal, which was provided by the versatile Großkreutz.
And despite their frantic victory, Dortmund actually qualified as group winners thanks to Arsenal's defeat in Italy.