Nottingham Forest moved into the top six thanks to a 1-0 win over Ipswich, while Newcastle were denied by Daniel Munoz's 94th minute equaliser for Crystal Palace in a 1-1 draw at Selhurst Park.
Arsenal can move into second if they avoid defeat at West Ham later.
Kluivert made history at Molineux to curtail Wolves' mini-revival.
The Dutchman opened the scoring, Jorgen Strand Larsen replied for Wolves before Milos Kerkez made it 2-1 for the visitors inside eight minutes.
Kluivert made it 3-1 just 10 minutes later and completed his hat-trick 16 minutes from time after Evanilson also set a record by winning a third penalty.
Strand Larsen pulled another goal back for Wolves, but Gary O'Neil's men slip back into the relegation zone.
Brentford continued their stunning home form to thrash the Foxes in front of Leicester's in-coming manager Ruud van Nistelrooy.
Facundo Buonanotte had put the visitors in front in west London.
But Brentford roared back to make it six wins and one draw from seven home league games this season.
Yoane Wissa started the comeback before Schade took centre stage.
The German had only scored two Premier League goals since joining the Bees in 2023 but matched that tally in 24 minutes to put the home side in front before half-time.
Schade then rounded off the scoring to show Van Nistelrooy the scale of the task that awaits him ahead of his first game in charge at home to West Ham on Tuesday.
Chris Wood's emphatically struck penalty was enough to get Forest back to winning ways and leave Ipswich still in the bottom three.
Palace edged out of the relegation zone on goal difference thanks to late drama against Newcastle.
Marc Guehi, who had been the subject of multiple bids from Newcastle during the transfer window, put the Magpies in front early in the second half with an own goal.
However, the England defender redeemed himself as from his cross Munoz powered home a header at the back post.
Runaway leaders Liverpool host Manchester City in a top-of-the-table clash on Sunday.
The Reds enjoy an eight-point lead over the defending champions, who are winless in six games.
Manchester United boss Ruben Amorim is also targeting his first Premier League win on Sunday when lowly Everton visit Old Trafford.
