The Spaniard powered home a sweet left-footed strike five minutes from time at the Red Bull Arena to breach Rangers’ stubborn defence.
“I just kicked the ball,” said the former Manchester City full-back. “It was a lot of team work until that point, but we were just missing a goal.”
Leipzig had struggled to break the deadlock as Rangers restricted them to just 10 shots despite boasting around 65 percent possession.
The Germans will be looking to reach their first European final — in Seville on May 18 against either Eintracht Frankfurt or West Ham — in next Thursday’s return game in Glasgow.
“It’s still very close, it will be very hard over there,” Angelino added. “We didn’t give them many chances and we deserved the win.”
However, this was a below-par display from the hosts, who had a 15-game unbeaten run ended last Saturday when they conceded two late goals at home to Union Berlin in the Bundesliga.
Leipzig coach Domenico Tedesco started with a three-man attack of Dani Olmo, Dominik Szoboszlai and Christopher Nkunku, with both Andre Silva and Yussuf Poulsen left on the bench.
Rangers’ back four sucked up everything the Leipzig attack threw at them early on, with Gers captain James Tavernier throwing himself into countless tackles.
A quarter of an hour had passed before the hosts created the first clear chance, but Nkunku could not get his shot away.
Both teams had opportunities early in the second half.
Ryan Kent fired just wide after dribbling into the area, then Tavernier had a shot charged down for the visitors.
Rangers goalkeeper Allan McGregor turned 40 in January, but he showed great reactions to block a powerful shot from Nkunku, who later fired wide with the goal at his mercy.
With 20 minutes left, Leipzig boss Tedesco swapped Olmo for Swedish playmaker Emil Forsberg.
USA midfielder Tyler Adams fired over just before Angelino settled the first leg when a corner was cleared to the wing-back, whose volley flew past McGregor.
Thousands of away fans travelled to Germany and Rangers manager Giovanni van Bronckhorst said the home support could help his team next week at Ibrox.
“Of course, you are always disappointed to concede a goal, we defended well and didn’t give a lot away,” said the Dutchman.
“The result could have been better so we know what we have to do next week.
“We have been behind before in a tie and it’s only a one-goal difference at home. We respect the opponent, but it always helps to play at home.”