Borussia Dortmund stormed into the Champions League semi-finals with a rip-roaring 4-2 home win over Atlético Madrid on Tuesday that saw them fight back from a 2-1 defeat in the first leg to win the tie 5-4 on aggregate.

The German side scored twice in the first half and looked to be cruising into the last four, but Atletico bounced back with two goals early in the second half to put Dortmund on the ropes before they dug deep to complete a superb comeback.

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Dortmund will now take on Paris Saint-Germain, who came back from 4-2 down to beat Barcelona 6-4 on aggregate, in the semifinals, with a place in the final on June 1 at London's Wembley Stadium at stake.

"We played an incredibly good first half, then there was the extremely unfortunate goal conceded, [but] how we came back, wow," delighted Dortmund coach Edin Terzic said in a TV interview.

The game got off to a flying start, and midfielder Marcel Sabitzer was only prevented from putting the home side ahead in the third minute by a last-ditch block from César Azpilicueta as the home side came flying out of the blocks.

Julian Brandt put Dortmund level on aggregate in the 34th minute, latching on to a brilliant pass from Mats Hummels and firing home, and five minutes later Dortmund took the lead in the quarter-final tie as Ian Maatsen cut in from the left and drilled a perfectly-placed shot in at the foot of the far post.

Atletico coach Diego Simeone made a triple substitution at the break and it paid dividends almost immediately as Dortmund defender Hummels turned the ball into his own net in the 49th minute, and substitute Ángel Correa put the Spaniards back in the lead in the tie in the 64th minute, scoring at the second attempt.

However, the home side struck back and striker Niclas Fullkrug made it 4-4 on aggregate with a brilliant glancing header from Sabitzer's cross in the 74th minute.

"I think about all the chances we created in the two games, but we weren't clinical," Simeone said. "We had the chance for [Samuel] Lino to make it 3-0 [in the first leg]. Congratulations to the opponent, we competed as well as we can, now we focus on LaLiga.

"They were better, and when a team is better you have to congratulate them."

The home fans had barely retaken their seats following that goal when Sabitzer popped up on the edge of the box, steadying himself with an excellent first touch before drilling home the winner in front of an ecstatic crowd three minutes later.

"It was a great game for everyone who came to the stadium. We played a great first half, it took us five or 10 minutes (to get going) in the second half, we have to accept that, but the answer to that [the Atletico goals] was sensational," goal-scorer Brandt said.