Jurgen Klopp's side fell behind in the 12th minute courtesy of a close-range header from Chiedozie Ogbene, but Luton's lead disappeared in a flash as the hosts turned the game on its head with two quickfire goals in the space of just 125 seconds midway through the second half.
Alexis Mac Allister assisted headers buried by Virgil van Dijk and Cody Gakpo, before Luis Diaz added a third on the 71-minute mark with a composed left-footed finish from inside the penalty area.
On his 100th appearance for Liverpool, Elliott put the game to bed with a superb first-time strike whipped into the top corner following a smart through-ball from 18-year-old debutant Jayden Danns to make sure Reds claimed three crucial points to boost their Premier League title hopes.
Klopp and co celebrated their 18th top-flight victory of the season and have consolidated their place at the summit, moving four points clear of reigning champions Manchester City in second as well as five points above Arsenal in third.
Elliott was handed just his fourth Premier League start of the season against Luton, replacing he injured Diogo Jota, and the versatile midfielder certainly took his chance in the first team as he posed a constant threat as one of the hosts' chief creators.
At the age of 20 years and 321 days, Elliott has become the youngest player on record (from 2008-09) to score, have five or more shots (five), create five or more chances (seven), have 10 or more touches in the opposition's box (17) and make at least 25 passes ending in the final third (27) in a Premier League
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