DORTMUND, Germany - Borussia Dortmund beat Olympiakos Piraeus 1-0 with a first-half goal from Kevin Grosskreutz on Tuesday to give themselves a Champions League lifeline and move up to third spot in Group F.

The Germany international stunned the visitors after seven minutes, rifling in from 22 metres for Dortmund's first Champions League win of the season which kept them in the running for a spot in the knockout stage.

The Greeks, needing at least a draw to seriously remain in contention for a top-two finish, battled hard but created few clear chances and it was Dortmund who could have scored again when Robert Lewandowski hit the post on the hour.

The result lifted the Bundesliga champions above the Greeks, into third place on four points from four games with Arsenal and Olympique Marseille leading the group on eight and seven points respectively following their 0-0 draw in London.

"There was a bit of luck in the game for us," said Dortmund coach Juergen Klopp, whose team travel to Arsenal next. "We started well but then our holding midfielders became a bit too attack-minded. But at the end of the day we deserved to win."

Olympiakos are on three and now need to win their last two matches to have any chance of advancing past the group stage.

Dortmund had described the game as a "final" and were eager to make amends for their 3-1 defeat in Athens last month.

They put the visitors on the back foot from the start with Ivan Perisic picking up a Lewandowski pass and firing just over the bar from inside the box after three minutes.

Grosskreutz missile

Grosskreutz, who had missed earlier, did it better four minutes later with Mario Goetze drawing two players on him and then laying the ball off for the midfielder who unleashed a blistering drive that left Olympiakos keeper Balasz Megyeri no chance on his Champions League debut.

The visitors gradually recovered and then threatened with a close-range effort from Francois Modesto and were unlucky not to be awarded a penalty for a hand ball by Dortmund central defender Mats Hummels just before the break.

In the second half, Olympiakos picked up where they had left off with Germany-born Jose Holebas testing Dortmund keeper Roman Weidenfeller with a powerful drive two minutes after the restart.

Dortmund could have added another goal with Lewandowski profiting from a bad Holebas back pass to beat Megyeri to the ball but hitting the post from a tight angle after an hour.

The Greeks almost snatched a last-gasp draw when Olof Mellberg was left unmarked in the box deep in stoppage time but headed over the bar.

"It was a divided game. We had our chances and we had them until the very last minute but we did not manage to score," Olympiakos coach Ernesto Valverde told reporters. "In the end we could not get the result we wanted and the early goal did not help."