Marcel Halstenberg's last-gasp penalty saw RB Leipzig snatch a 2-1 victory over Augsburg in a nerve-shattering German Cup quarter-final tie on Tuesday.

After Alfred Finnbogason's equaliser had rescued Augsburg in injury time of normal time, Halstenberg restored the lead in injury time of extra time to send Leipzig into the first German Cup semi-final of their history.

The hosts held Leipzig at bay for long periods and looked dangerous themselves until a momentary lapse in concentration allowed Werner to break away and slot in the opener on 74 minutes.

Leipzig were left shellshocked, however, when Finnbogason snatched an equaliser four minutes into injury time.

Yet it was Leipzig who had the last laugh in a jaw-dropping game, as Michael Gregoritsch's late handball in the box broke Augsburg hearts and gifted Leipzig victory.

"We are so frustrated because we were at least on the same level as Leipzig today," Augsburg coach Manuel Baum told Sky.

Tensions bubbled over after full-time, as the two teams' sporting directors Oliver Mintzlaff and Stefan Reuter clashed on the sidelines.

"It was a very emotional game, it was a fight, and tempers flared a bit at the end," Mintzlaff told ARD.

"The team fought so hard, and we deserved to win."

Leipzig will be joined in the last four by Hamburg, who beat Paderborn 2-0 earlier in the evening.

Hamburg striker Pierre-Michel Lasogga continued his fine scoring form in the German Cup as he grabbed both goals to fire his side into the semi-finals for the first time in ten years.

Lasogga has now scored six goals in the competition this season, equalling a club record set by Ivica Olic the last time Hamburg reached the last four in 2009.

"I've never been in the semi-finals before, and I can't describe how good it feels," Lasogga told Sky.

"We deserved to win today. It was an extraordinary evening."

After a handful of chances after half-time, Lasogga gave Hamburg the lead on 54 minutes, rising high at a corner to nod the ball in from close range. Fourteen minutes later, he smashed a powerful shot into the bottom corner to seal Hamburg's passage to the semi-finals.

The other two semi-finalists will be decided on Wednesday, as Bayern Munich host Heidenheim and Werder Bremen travel to Schalke.

The draw for the semi-final will take place next Sunday.