Villa were ahead after just 153 seconds thanks to John McGinn, but Leipzig levelled things up midway through the first half when Lois Openda exploited a mix-up in defence.
Jhon Duran's 53rd-minute stunner was cancelled out by a Christoph Baumgartner goal nine minutes later, setting the stage for the substitute Barkley to blast in just two minutes after coming on.
The goal, a deflected long-range effort, was Barkley's first in the Champions League and has Villa on track for a top-eight finish in their return to Europe's top competition after 41 years.
McGinn told the UEFA website the goal "felt amazing, because it's been a bit of a dry spell for me.
"I think over 90 minutes we were the better team -- Leipzig's goals were against the run of play."
"That's the role of the lads on the bench," Barkley told Amazon Prime. "I'm happy to come off the bench and score the winner."
"They all count, don't they?" Barkley said of his deflected winner.
Teams finishing in the top eight qualify directly and avoid an additional set of knockout fixtures.
The loss, Leipzig's sixth in six Champions League games this season, extinguished the hosts slim hopes of a place in the knockouts.
A "disappointed" Openda told reporters his side needed to "keep fighting... This can happen. We will get another chance for next year and we will need to learn from our mistakes."
Villa travelled to Germany having won two straight Premier League games to break an eight-game winless run.
Leipzig however were very much at the last-chance saloon, with five losses in five putting them on track to miss the knockouts -- a fate they only suffered twice in six seasons under the previous format.
Returning to his preferred European stage, Villa manager Unai Emery urged his side to "break our limit" and the visitors heeded his words early, opening at break-neck speed.
McGinn put Villa ahead three minutes in, hammering home from close range after Ollie Watkins nodded a perfect header into his path.
McGinn's goal was just one of many chances inside the opening 15 minutes, including golden opportunities for Watkins and Youri Tielemans as Villa's movement and precision overwhelmed the sluggish hosts.
Leipzig's young side have lacked confidence this season, particularly in Europe, so they were fortunate when a poor defensive mix-up allowed Openda to level things up after 27 minutes.
Villa's centre-backs and goalkeeper Emiliano Martinez held off a high ball, allowing Openda to slip through, collect it before the bounce and slot calmly into an empty net.
With Leipzig in the ascendency as half-time came, Emery summoned John Duran from the bench at the break and the Colombian had Villa back in front with an incredible goal eight minutes in.
Fed by Tielemans on the break, Duran collected the pass and unleashed a dipping shot from well beyond the penalty arc which evaded Leipzig goalie Peter Gulacsi.
Duran looked to have scored again minutes later but his strike was ruled out for the narrowest of offsides by VAR.
The hosts hit back seconds later, Openda breaking through on the counter and chipping for Baumgartner, who volleyed Leipzig level.
Summer arrival Barkley was the visitors' unlikely hero, his deflected shot -- the 31-year-old's first Champions League goal -- finding the net and continuing Villa's incredible return to the European stage.
