Two first-half goals from Marcel Sabitzer secured the German side's last-eight place.
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Marcel Sabitzer struck twice inside the opening 21 minutes as Leipzig booked a place in the quarter-finals for the first time.
The match at a glance
Home skipper Sabitzer quickly dispelled any notion of nerves from the UEFA Champions League knockout debutants with two early goals in Germany. The Austrian doubled Leipzig's aggregate lead after just ten minutes, a speculative low shot from the edge of the box sneaking inside Hugo Lloris' right-hand post via his fingertips.
The French goalkeeper, man of the match in the first leg three weeks ago, perhaps could have done better. He certainly should have 11 minutes later when Sabitzer's near-post header found the net following Angeliño's cut-back. There was no way back for Tottenham, Dele Alli passing up their best opening. Substitute Emil Forsberg made it 3-0 late on with his first touch.
Man of the match: Marcel Sabitzer (Leipzig)
Roberto Martínez, UEFA technical observer: "A match-winning performance from the midfielder, with two clinical finishes and a commanding display. Alongside Konrad Laimer, he controlled the midfield area and that was the difference between the teams on the night."
Home view: James Thorogood, Leipzig reporter
Leipzig's composure over the course of the tie against last season's runners-up belied their status as knockout stage debutants. Here they produced answers to the questions posed, all the while proving too great a conundrum for a depleted Tottenham to solve. Julian Nagelsmann becomes the youngest head coach to reach the UEFA Champions League quarter-finals, but most pleasing will be the levels of maturity and confidence his side maintained for 180 minutes.
Visitors' view: Mark Pettit, Tottenham reporter
Tottenham's defensive frailties, highlighted by José Mourinho yesterday, were laid bare against a purposeful Leipzig team and this tie was as good as over inside 21 minutes tonight. Given Spurs' league position and form it is highly unlikely they will back on this stage next term, making this a distinctly underwhelming end to four consecutive seasons of UEFA Champions League football that have produced some of the most incredible scenes in the club's history.
Julian Nagelsmann, Leipzig coach: "I'm totally happy because we're through to the next round, and I'm totally happy because, in the end, it’s fully deserved. We scored four, conceded none, were the better team than Tottenham and we were the better team today."
Angeliño, Leipzig defender: "It’s very special. A great night and I’m over the moon right now. We were very positive and went for it straight away. We’re really happy. It wasn’t easy – we had to work very hard. The plan was clear on the pitch. We were dominant but we worked really hard."
José Mourinho, Tottenham manager: "I was very positive yesterday and have to be all the time. We all believed. It is difficult, they are a very strong side. Our first couple of mistakes and they score, then the game is very difficult. Their physicality is very good. Midfield very intense, attack very fast."
Dele Alli, Tottenham midfielder: "It's devastating. We had belief before the game but didn’t perform. If you don’t step up you will be punished and it’s disappointing to go out like that. After the first went in, it didn’t change much as we needed two anyway. We conceded too many chances as a team and we were punished. It knocks the stuffing out of you."
- Leipzig are the seventh German club to reach the quarter-finals in the UEFA Champions League era after Bayern, Dortmund, Leverkusen, Kaiserslautern, Schalke and Wolfsburg.
- Sabitzer has scored in 15 matches for club and country this season and his side have won all of them.
- At 32, Nagelsmann is the youngest coach to win a UEFA Champions League knockout tie, breaking a record previously held by Didier Deschamps, who was 35 when he led Monaco to the final in 2003/04.
- Leipzig are the first team in 15 attempts to deny Tottenham an away goal in the UEFA Champions League.
- All four of Spurs' UEFA Champions League defeats this season came against German outfits (two against Bayern, two against Leipzig). Only one side has ever lost more in a single campaign against teams of the same nation: Leeds United were defeated five times by Spanish opponents in 2000/01.
Leipzig: Gulácsi; Klostermann, Upamecano, Halstenberg; Mukiele (Adams 56), Laimer, Sabitzer (Forsberg 87), Angeliño; Nkunku (Haidara 59), Schick, Werner
Tottenham: Lloris; Alderweireld, Dier, Tanganga; Aurier (Fagan-Walcott 90+1), Winks, Lo Celso (Gedson Fernandes 80), Sessegnon; Lamela, Alli, Moura