Domenico Tedesco's men were the aggressors for much of the game, but the Bundesliga side struggled to create chances against a resolute Gers backline, until the former Manchester City left-back's strike with five minutes remaining.
Leipzig dangerman Christopher Nkunku – who has 30 goals and 16 assists in all competitions this season – was unable to add to his tally, with the visitors keeping the Frenchman relatively quiet at Red Bull Arena.
It was not quite the famous night in Germany that Rangers experienced earlier in the campaign when they beat Borussia Dortmund 4-2 at Signal Iduna Park, but with just a one-goal deficit, it leaves the tie open ahead of the second leg at Ibrox next week.
Despite a dominant start, the closest Leipzig came in the first half was when Konrad Laimer managed to break down the left side of the Rangers penalty area, but his shot was well blocked by Connor Goldson.
The Scottish side were happy to sit deep and try and release Ryan Kent and Scott Wright with long balls, which could explain why their pass completion percentage in the first half was just 66.9.
Giovanni Van Bronckhorst's side had their first real chance early in the second half when Kent received the ball in space on the left, before beating his man on the outside and firing a shot across Peter Gulacsi's goal, but wide.
The first shot on target arrived in the 53rd minute as Nkunku hit a strong effort from just outside the box, which Allan McGregor palmed away, with the former Paris Saint-Germain attacker then somehow missing the target after rounding McGregor with 20 minutes to go.
The deadlock was finally broken in the 85th minute when a corner from Nkunku was headed away by Fashion Sakala, only for Angelino to volley the ball back through everyone and past the despairing dive of McGregor as the German side take a slender lead to Scotland.