Angeliño scored five minutes from time to finally break Rangers' resistance in a keenly fought semi-final first leg.
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A late volley from Angeliño secured victory for Leipzig and a 1-0 advantage against a stubborn Rangers side at the halfway stage of their UEFA Europa League semi-final.
48' Kent slides shot across face of goal
53' McGregor saves Nkunku effort
69' Nkunku misses golden chance
85' Angeliño volleys in only goal
Match in brief: Angeliño strike nudges Leipzig ahead
Leipzig had more of the ball in the first half, but their only chance fell to Christopher Nkunku, who was unable to get his shot on target.
The game opened up more after the interval and Ryan Kent sent an effort across the face of goal, before Nkunku's well-struck shot was pushed away by Allan McGregor.
Nkunku then spurned a great chance to put his side ahead, running through the defence and getting past McGregor, only to send his effort high and wide of the unguarded target.
But just as it looked as though the match would end in a stalemate, the deadlock was broken when Leipzig's only corner of the night was headed out as far as Angeliño, who returned it with considerable interest, out of McGregor's reach and into the back of the net.
James Thorogood, Leipzig reporter
If this tie against Rangers stands as a test of Leipzig's title-winning credentials, they passed the first leg, albeit without flying colours. The hosts were more proactive across the 90 minutes and, while they struggled to find solutions to the problems Rangers posed, head coach Domenico Tedesco has a one-goal lead to work with ahead of a trip to Ibrox that remains just as tricky even with an advantage.
Alex O'Henley, Rangers reporter
With five minutes remaining, Rangers were looking at a defensive masterclass until Angeliño's volley from distance broke the deadlock. It's still a decent result for Rangers away from home, but they will have to be more expansive in the return leg at Ibrox next week. You suspect a more open game might also suit the Bundesliga side, who didn't really show their quality until Angeliño's late strike. It's all still to play for with the prize of next month's final very much up for grabs.
Joe Aribo, Rangers forward, speaking to BT Sport: "All in all, we are disappointed. We thought we could see the game out, but there's still a big game to come at Ibrox. We wanted to stay compact and we did that for the most part until they got that goal late in the game."
Tyler Adams, Leipzig midfielder: "It was one of those games where we had to dig deep. They had a clear game plan and made life hard, but we got the result we wanted. It was a wonderful goal from Angeliño."
Giovanni van Bronckhorst, Rangers manager speaking to BT Sport: "It's always disappointing when you lose a goal so late in the game, but it's a 1-0 defeat. The players did well and we know what we have to do next week. We play at home and the crowd at Ibrox will give us the energy we need."
Kenny Miller, BT Sport
"It's not a disaster. It's not a bad result to take back to Ibrox. The tie is very much alive and it's all to play for next week."
- Leipzig have lost only two of their 11 UEFA Europa League knockout phase matches (W6 D3).
- Leipzig ended a run of three successive home draws in European competition.
- Rangers failed to score for only the seventh time in all competitions this season – five of those games having come in Europe.
- Rangers have won just one of their nine away matches in European competition this season (D3 L5).
- Rangers have only won one away game in the semi-finals of a European competition in their history.
Leipzig: Gulácsi; Gvardiol, Henrichs (Mukiele 89), Halstenberg; Laimer, Adams, Klostermann, Angeliño; Olmo (Forsberg 71), Nkunku (Poulsen 89), Szoboszlai (André Silva 71)
Rangers: McGregor; Tavernier, Goldson, Bassey, Barišić; Jack (Sands 83), Lundstram; Aribo (Arfield 83), Kamara, Kent; Wright (Sakala 69)