If Rangers are to reach their first European final in 14 years, they will have to come from behind.
The hosts were limited in their opportunities but took a slender lead to Glasgow thanks to a stunning Angelino volley five minutes from time.
Ryan Kent had a rare chance for Rangers, but he missed the target.
Their performance in Germany was composed, with a vocal home crowd growing frustrated as time ebbed away before a goal that illuminated an otherwise tense tie.
Rangers are hoping to get a rowdy Ibrox next Thursday as a night of drama is about to start in Govan.
On a warm night in eastern Germany, there was a sense of familiarity with the local brew and revelry.
The road Walter Smith's team traveled to Manchester in 2008 is remembered for some of the most disciplined defense you're likely to see. The teams that will testify to that are Panathinaikos, Werder Bremen, Sporting Lisbon and Fiorentina.
In the Red Bull Arena, it took a moment of brilliance in the 85th minute to deny Rangers another clean sheet.
As Fashion Sakala's headed clearance descended towards Angelino on the edge of the box, the Spaniard didn't break stride to meet the ball perfectly. The din was replaced by a quiet before he shot his arrow past Allan and into the far corner.
The flow of traffic towards the goal was as relentless as the chanting, singing and bellowing from inside the stadium, and it was hard to stomach given Rangers' composure and improved performance after the break.
Christopher Nkunku was the first to be frustrated after Calvin Bassey denied him. Konrad Laimer would have had a thundering effort if it hadn't been for the quick reactions ofConnor Goldson. There was a block near the line.
Rangers fans had little to shout about for most of the first half, but they gave it a go anyway. Kent was isolated while Scott Wright was nervous.
The restart roused a focus from the visitors.
A quick break was led by the scampering Kent, with Joe Aribo now playing through the middle. The Englishman rifled the ball across goal after Lukas Klostermann was left flustered by one jink and duke after another. The Rangers fans in the far corner were in disbelief.
The venue, crowd and occasion deserved were all threatened by the game. The rattled hosts replied instantly after Nkunku rammed a right-foot shot at goal.
Nkunku would desert him soon. The gifted Frenchman, who has 30 goals this season, snuck past John Lundstram and rounded the corner but, with the goal gaping, hoisted an off-balance shot high and wide. It might still be in the night sky.
Angelino's moment of class brought bedlam to the Red Bull Arena as the intense pressure on the Rangers defence increased.
Rangers in Europe can be fun to watch. Over the 17 games they have played, their mission to frustrate and frustrate an astute Bundesliga side seeking their first major trophy almost worked to perfection. It took a goal worthy of the occasion to ruin that plan.
We have been here before with the Scottish champion.
The first leg of the quarter-finals was won bySporting Braga, but they were hit with a completely different beast when the tie got back to Glasgow. That will give a lot of belief that the tie can be salvaged.
The dilemma of who plays up front still exists. Wright struggled at the top of the attack in the first half, while Aribo looked more comfortable there after the break. If Kemar Roofe is still out, will he lead the line next week?
It was a tough game. Rangers are not that deep and it was hard to play against them.
In the second half, we had three or four good chances and the most difficult one to score. We know thatAngelino would shoot 10 times there and score twice.
Rangers have to come out more, maybe not from the start, and when there are more spaces we are better.
We were well-organised. The first half was more about making sure they couldn't create goals. In the second half, we were more confident on the ball.
It is always disappointing when you lose a goal late in the game. The players played well and next week we will be at home and Ibrox will give us the energy we need.
We need to attack them with more bodies. We can overcome a 0-1 defeat.
The Rangers have big games coming up. The final Old Firm game of the season will be played at Celtic Park on Sunday, before the team goes to Glasgow to face Leipzig on Thursday.
Formation 3-4-1-2.
16Klostermann32Gvardiol23Halstenberg39Henrichs27Laimer14Adams3AngelinoFormation 5-3-2.
2Tavernier6Goldson4Lundstram3Bassey31Barisic8Jack17Ayodele-Aribo18KamaraThe match is over, Rangers 0.
The second half ended with a score of 1, Rangers 0.
The corner is for RB Leipzig. James Tavernier was Conceded.
James Tavernier was shown a yellow card for a bad foul.
The left wing is where Yussuf Poulsen won the free kick.
James Tavernier (Rangers).
The yellow card is shown toConnor Goldson.
Silva won a free kick in the defensive half.
The foul was committed by the Rangers'Connor Goldson.
Foul by Forsberg.
Borna Barisic won a free kick.
There was a substitution, RB Leipzig. Benjamin Henrichs was replaced by Nordi Mukiele.
There was a substitution, RB Leipzig. Christopher Nkunku was replaced by Yussuf Poulsen.
Foul by Angeli.
James Tavernier won a free kick.
Goal! Rangers were blanked by RB Leipzig. Angeli shot from outside the box to the bottom left corner following a corner.
The corner is for RB Leipzig. They had a match with Allan McGregor.
The attempt was saved. Tyler Adams' shot from outside the box was saved in the bottom right corner. Christopher Nkunku assisted.
The Rangers had a substitution. Scott Arfield is the new person.
The Rangers had a substitution. Ryan Jack was replaced by James Sands.
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