After Dimitri Payet was sent off the German side missed chance after chance, before Grosskreutz finally settled the game with just three minutes remaining.
Here, Sports Mole analyses whether the outcome of Group F was fair.
On target 4
On target 11
Was the result fair?
Definitely. After Payet was sent off in the first half there was only one side likely to score, but even so it did not look like it was happening. Both Lewandowski and Marco Reus missed glorious chances before Grosskreutz finally scored. On another day it could have been 5-1, but in the end Dortmund did just about all that they needed to in order to qualify.
The sending-off was the turning point, as before that it was quite an even game. They got off to a slow start but equalised after a bit of pressure. Once they were down to 10 men the plan was to defend and hope for the draw. For a while it worked, but they eventually dropped so deep that many chances were conceded. Dortmund needed just one and took it in the end.
They were good in most areas apart from their finishing. Roman Weidenfeller's part in the Marseille goal might worry Jurgen Klopp, but his side were comfortably on top in all areas. They certainly created enough chances to win two games of football, but they deserve a lot of credit for sticking to the plan and finally getting the late winner.
Sports Mole's man of the match
Kevin Grosskreutz: He scored the winner, but even before that he was up and down the flank for the full 90 minutes. He provided the width on the right which allowed the Dortmund attacking trio to rotate but still always have the option to pass wide. His performance was deserving of the goal and as a Dortmund boy he will be more delighted than anyone else with the win.
Had Dortmund not managed the win, this could go to either Lewandowski or Reus for their misses. Instead, Payet's dreadful attempt at a dive wins the award for Marseille. He had just been booked and after a tackle in which the ball was won, he decided to hurl himself on the ground but did not fool the referee. He was quickly shown his second yellow and sent from the field.
Marijo Strahonja got one big decision wrong, but in the end it did not make a difference to the result. Just before Grosskreutz's winner, Mandanda fouled Lewandowski in the box, but Strahonja waved play on. Apart from that he got all the decisions right, in particular the one to show Payet a second yellow for a dive.
Marseille: The search for a new coach goes on, but the players are back in Ligue 1 action on Sunday. They face the tough trip to Lyon as they look to get back into the top five.
Dortmund: Dortmund, boosted by this win, will travel to Hoffenheim to try to end a poor run of Bundesliga results, having lost three of their last four.