Competing in the Europa League 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 Lukas Haraslin 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 Nunez’s strike from range deceived Jensen.
The contest remained incredibly competitive as the first half progressed, with Sparta’s Veljko Birmancevic 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 half time, as Nunez 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 Birmancevic’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 Diaz’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 Jurgen 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 Europa League without a win.
Flashscore Man of the Match: Darwin Nunez (Liverpool)
Bayer tested as Roma batter Brighton
Elsewhere in the Europa League's other last-16 early kick-offs, Azerbaijan's Qarabag almost stunned Bayer Leverkusen, who are still unbeaten this season.
The hosts were leading the match 2-0 in Baku before Bayer pulled it back, levelling the scores in the final moments thanks to a Patrik Schick strike.
Read a full match report here.
Over in Rome, resurgent AS Roma emphatically defeated out-of-form Premier League side Brighton 4-0 in their first leg to continue their fantastic run under recently-appointed coach Daniele De Rossi. Brighton will need a miracle to turn the tie around next week.
Read all about Roma's big win here.
