After struggling to offer a meaningful attacking threat in the first half, Real Madrid found their cutting edge within three minutes of the restart when Brahim produced a sublime solo effort to score his eighth competitive goal of the season.
With a narrow lead to protect, it was left up to Andriy Lunin to pull off a number of crucial saves to ensure Los Blancos returned to the Spanish capital with a first-leg advantage.
Real Madrid travelled to Germany in high spirits after recording a dominant 4-0 win over title rivals Girona in Saturday's La Liga contest.
Los Blancos were rarely tested by Michel's side, but they were put under pressure in the early stages at the Red Bull Arena.
Benjamin Sesko briefly thought he had given the hosts a second-minute lead, only to see his header ruled out after the officials ruled that the offside Benjamin Henrichs was interfering with play.
The 20-year-old was presented with another promising opportunity when he latched onto Dani Olmo's neat through ball, but he could only fire his shot straight at the resolute Lunin.
With Real Madrid unable to call upon the injured Jude Bellingham, they struggled to find a cutting edge in a first period where Leipzig carried the greater threat.
However, that quickly changed after the half-time break when Brahim produced a sensational solo effort to give Real Madrid a precious lead.
The 24-year-old, who replaced Bellingham in the lineup, skipped inside from the right flank before he sent an unstoppable left-foot strike out of Peter Gulacsi's reach and into the far corner.
At the opposite end of the pitch, Lunin was required to make an important double to stop Olmo and Sesko from restoring parity.
As the contest reached the 70-minute mark, Lunin made another key save to deny Olmo and kickstart a Real Madrid counter-attack.
Within seconds, Vinicius Junior worked the space to send a prodded effort towards goal, only to see his strike bounce off the far post.
As Leipzig stepped up their efforts to find a late leveller, Lunin showed his worth to Carlo Ancelotti's side, pulling off two saves in quick succession to keep out Xavi Simons and Sesko.
Just moments later, the Ukrainian shot-stopper pushed Amadou Haidara's strike away from danger to frustrate the hosts and effectively seal the first-leg victory.
After suffering their fourth defeat in the last six games, Leipzig will have significant work to do when they travel to the Santiago Bernabeu at the start of March.