Jurgen Klopp's side reached the last-16 by their skin of their teeth thanks to the latest of goals in France
Last season's beaten finalists looked to be on their way to a routine victory when Robert Lewandowski put them ahead in the opening stages.
However, Marseille - led by sporting director Jose Anigo following Elie Baup's dismissal last week - quickly levelled matters through defender Souleymane Diawara as they sought to avoid the ignominy of finishing the group stage with no points.
Dortmund wasted numerous chances to go back in front in a one-sided second half at the Stade Velodrome, with Lewandowski and Marco Reus among those to miss gilt-edged opportunities.
But the Bundesliga club finally breached a stout Marseille defence as Grosskreutz struck three minutes from time to seal top spot as earlier group leaders Arsenal were beaten 2-0 at Napoli.
Defender Rod Fanni made his return from injury for Marseille, while Benjamin Mendy and Saber Khelifa were also included in the starting XI by Anigo.
Nuri Sahin recovered from an ankle problem to feature in the Dortmund line-up as Klopp handed starting berths to youngsters Marian Sarr and Erik Durm.
Dortmund took the lead after just three minutes, left-back Durm playing an influential role as he played in Lewandowski with a deft pass, with the Poland striker making no mistake as he coolly lifted the ball over Steve Mandanda.
However, the visitors' hopes of qualification were dealt a blow 11 minutes later when Diawara headed into an empty net after Khelifa had turned Dimitri Payet's superb free-kick against the crossbar.
Dortmund continued to dictate an extremely open game, though, but struggled to create many meaningful chances. Reus went closest just after the half-hour mark when he blazed over the bar following a swift counter-attack.
But the German outfit's fortunes turned shortly after as Payet was dismissed for a second booking in a matter of minutes after referee Marijo Strahonja ruled that the winger had dived in an attempt to win a penalty from a challenge with Sahin.
Marseille remained a threat despite their numerical disadvantage, however, with Andre-Pierre Gignac forcing Roman Weidenfeller into action before Reus spurned another opportunity at the other end prior to the interval.
Anigo's men rode their luck for much of the second period, and they were fortunate not to fall behind on two separate occasions, Reus hitting the post shortly after Mandanda had produced a brilliant far-post save to keep out Jakub Blaszczykowski's header.
Dortmund should have re-taken the lead in bizarre fashion after 66 minutes when Lewandowski intercepted Mendy's backpass, only to hit the side netting after rounding Mandanda.
Klopp's men refused to let up as they heaped further pressure on the Marseille defence, Reus somehow missing the target from point-blank range with the goal at his mercy.
And Dortmund were then denied a penalty in the closing stages after Mandanda collided with Lewandowski, but this proved immaterial as Grosskreutz's effort from the edge of the area snuck into the bottom corner to ensure their progression.