The margin for error was non-existent for Leipzig, and a depleted home crowd were on their feet inside the opening minute when Antonio Nusa slotted through Benjamin Sesko, only for the Slovenian to fire high and wide.
Things quickly soured from there, as after just 153 seconds of action Villa had taken the lead in sumptuous fashion.
A probing move found its way to Matty Cash on the right wing, with his cross nodded down by Ollie Watkins into the path of the untracked John McGinn, who made no mistake in slotting past Peter Gulacsi.
Unai Emery’s men only seemed to grow in confidence from there, moving the ball nicely and creating openings for a second, though neither Watkins nor Youri Tielemans made the most of promising openings.
Villa were in complete control of the contest, but from nowhere, an Emiliano Martínez mistake handed Leipzig a route back into the contest.
Nicolas Seiwald’s long ball forward looked innocuous, but having come to collect it, Martinez backed out at the last minute, allowing Lois Openda to steal in and convert his first UCL goal of the season.
In response, Emery replaced the out-of-sorts Watkins with super-sub Jhon Duran, who needed just seven minutes to make his mark on this contest.
Left in space from 25 yards, the Colombian unleashed an audacious curling strike that looped over Gulacsi and restored the visitors’ advantage.
Duran had the ball in the net again six minutes later, only to be denied by an offside against Cash in the build-up.
That reprieve proved crucial for the Red Bulls, who levelled the contest for a second time against the run of play just after the hour.
Once more caught out by a long ball, the Villa defence failed to deal with Openda, who held the ball up and crossed for the arriving Christoph Baumgartner to arch a finish into the far corner.
Villa’s defensive calamities nearly saw them fall to defeat in the final stages as Openda seized on Pau Torres’ dreadful square pass, but the Belgian failed to convert a huge chance to earn victory for the hosts.
That’s a miss that would cost Marco Rose’s men dearly, as substitute Ross Barkley’s potshot from the edge of the box took a wicked deflection off Lukas Klostermann on its way in.
The result puts an end to Villa’s four-match losing streak on the road and lifts them up to third in the UCL standings, but another defeat and elimination for Leipzig surely puts Rose’s job on the line.