Competing in the UEL round of 16 for the first time since 2015/16, Sparta knew a fast start was needed in front of a raucous home crowd.
It was Liverpool, however, who came flying out the blocks, and their early pressure was rewarded inside the opening five minutes when Alexis Mac Allister was tripped inside the area by Asger Sorensen.
The Argentine assumed the responsibility from the spot and sent Peter Vindahl Jensen the wrong way to give the visitors a crucial lead.
Undeterred by that setback, Sparta came agonisingly close to a swift response when Lukáš Haraslín saw a close-range effort superbly thwarted by Caoimhin Kelleher.
Brian Priske’s men continued to probe for an equaliser, but despite plenty of encouraging play in the final third, Liverpool soon doubled their advantage in the 25th minute as Darwin Núñez’s strike from range deceived Jensen.
The contest remained incredibly competitive as the first half progressed, with Sparta’s Veljko Birmančević failing to sort his feet out in time to convert into an empty net, while at the other end, Cody Gakpo saw a pair of efforts expertly thwarted by Jensen.
Both sides continued to create chances almost at will, and Liverpool duly added a third on the stroke of HT, as Núñez rifled a sublime half-volley into the far corner.
Faced with a daunting three-goal deficit, the hosts pulled one back within a minute of the restart when Birmančević’s low delivery was sliced into his own net by substitute Conor Bradley.
However, the visitors swiftly fired back with a goal of their own, as Luis Díaz’s close-range strike was deflected beyond a helpless Jensen.
After a frantic first hour, the match settled into a more sedate rhythm heading into the closing stages, with Jürgen Klopp giving Dominik Szoboszlai and Mohamed Salah crucial minutes ahead of a crunch clash against Manchester City at the weekend.
There was still time, however, for the Hungarian to tuck home a fifth in stoppage time as Liverpool emphatically ended a run of 11 away knockout matches in the UEL without a win.
Flashscore Man of the Match: Darwin Núñez (Liverpool)
