Jurgen Klopp's team opened up a four-point advantage at the top of the table by dismantling the Hatters, who had initially gone ahead through Chiedozie Ogbene's finish in the first half.
The Hatters clung onto their advantage for the remainder of the opening period, but with Klopp's men attacking the Kop end during the second 45, Van Dijk sparked the opening of the floodgates.
The centre-back got Liverpool back on terms with a fine header from Alexis Mac Allister's corner, marking his 18th Premier League goal for the Reds since the start of the 2018-19 campaign.
Twelve of those efforts have come on familiar territory, seeing Van Dijk score more Premier League goals for Liverpool at Anfield than any other defender in the club's history.
The previous record holder was Finnish centre-back Hyypia, who spent 10 years on Merseyside from 1999 to 2009 and notched 11 top-flight goals at Anfield during his playing heyday.
Van Dijk had previously netted at Anfield this season in January's 5-2 FA Cup thrashing of Norwich City, but Wednesday's header marked his first Premier League home goal of the season and his second strike in the 2023-24 top flight overall.
The 32-year-old opened his account for the Premier League season away to Sheffield United in December, and his most recent Anfield top-flight goal came in a 2-0 win over Wolverhampton Wanderers in March 2023.
Following Van Dijk's equaliser, another cross from Mac Allister did the damage just three minutes later, as Luton were caught unawares from a throw-in and fell behind as Cody Gakpo nodded in from the Argentine's
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