Darwin Nunez struck twice following an Alexis Mac Allister penalty during a chaotic 45 minutes, in which the 3-0 scoreline certainly did not tell the full story, as the Maroons wasted a glut of openings and could have easily struck three or more themselves.
Conor Bradley's own goal right at the start of the second half gave Sparta a crumb of hope, but Luis Diaz and Dominik Szoboszlai killed that hope, putting the visitors - who ended an 11-game winless streak in Europa League knockout away games - in complete control ahead of next week's return fixture at Anfield.
Despite the impending visit of Premier League title rivals Manchester City, Jurgen Klopp went strong on the Prague soil, where Nunez, Cody Gakpo and Diaz formed a formidable front three with Mohamed Salah on the bench.
Nunez could have sent the Reds ahead with just two minutes gone, only to see his tame effort cleared off the line, but Klopp would not have to wait too much longer for his side's first goal of the knockout rounds.
After being felled by Asger Sorensen inside the penalty box, Mac Allister dusted himself down and calmly sent Peter Vindahl Jensen the wrong way from 12 yards to propel Liverpool into a sixth-minute lead.
However, the visitors had only awoken a Sparta beast by scoring first and soon came under incessant pressure against their Czech hosts, whose winger Lukas Haraslin came within inches of equalising in the 15th minute, but his back-post shot was palmed into the air by Caoimhin Kelleher before Joe Gomez produced a spectacular goal-line clearance.
Sparta continued to pepper the Liverpool goal with attempts - Ibrahima Konate came up with a crucial block to deny Ayrton Preciado in the 20th minute - one moment before Kelleher tipped a Martin Vitik attempt to safety.
Having weathered a category five Sparta storm, Liverpool surged down the other end of the field and doubled their tally entirely against the run of play in the 25th minute, as Nunez's swerving shot from range beat Jensen all ends up and flew into the centre of the goal.
Vindahl Jensen did not cover himself in glory, although neither were his attackers, as Brian Priske's side wasted another glorious opportunity in the 37th minute - Veljko Birmancevic somehow shot wide with the goal gaping after Kelleher denied Jan Kuchta.
Sparta certainly left themselves vulnerable to being punished by their visitors, and true enough, Liverpool's lead increased to three in first-half injury time, as Nunez met Mac Allister's ball over the top and found the far side of the net with aplomb, bringing up Liverpool's 1,000th competitive goal under Klopp.
The ball would finally end up in the back of Kelleher's net within just 39 seconds of the restart, although it was a Liverpool player who ironically had the telling contribution, as Bradley - only just on for Gomez - directed Birmancevic's cross into the top corner of his own goal; an excellent finish, albeit at the wrong end of the pitch.
After receiving a lifeline, Sparta will have also been encouraged to see Konate go off injured after 50 minutes - although Virgil van Dijk was an intimidating replacement - and Klopp's side restored their three-goal cushion just three moments later.
A patient build-up sequence down the right saw Bradley play the ball inside to Harvey Elliott, who in turn teed up Diaz inside the box, and the Colombian's low strike deflected off the leg of Vitik and sailed over a helpless Vindahl Jensen.
As the clock ticked down, Klopp introduced Salah for his comeback appearance - allowing the winger to get vital minutes into his legs ahead of the weekend - and the Egyptian briefly thought that he had capped off a five-star Liverpool display in the 85th minute.
Salah nonchalantly rolled the ball into the far side of the net from Szoboszlai's pass, but following a VAR review, the 31-year-old's goal was chalked off for offside - replays seemed to show that he was at least level with the last defender, though.
However, the fifth would arrive for Liverpool with the last attack of the game, as Szoboszlai was given the freedom of the middle third to charge at the Sparta backline and saw his strike fly into the top of the net via a nick off of Vindahl Jensen's outstretched leg.
Prior to next Thursday's second leg at Anfield, Liverpool host Man City in Sunday's top-of-the-table Premier League battle, while Sparta continue their bid for Czech Fortuna Liga glory away to Viktoria Plzen on the same afternoon.