Aguero got the slightest of touches on to Kevin De Bruyne's left-wing cross to open the scoring for City before half-time.
Within moments the visitors were back in the game as Clark seized on a mistake by Nicolas Otamendi and released Murphy with a precise pass. Murphy raced clear of Zinchenko and calmly finished with a neat dink over Ederson. One of the finest pieces of skill of the season so far to run and weave through to set up Aguero's hat-trick goal.
"We needed it (this win) after last week", Aguero told BT Sport. "It's a long season though and we've not won anything yet". Tottenham's Harry Kane (-185) is the homegrown favorite, but he has only one more goal than Liverpool's Mohamed Salah (+400), who at 4-to-1 offers enticing value.
City have had a mixed week off the pitch, having chose to end their attempts to sign Arsenal forward Alexis Sanchez, but securing central defender Nicolas Otamendi on a contract extension that runs until 2022.
With Sanchez inching towards a £35 million (39.6 million euros, $48.5 million) transfer to United rather than City, Gunners manager Arsene Wenger left him out of his matchday squad.
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At Turf Moor, United striker Anthony Martial scored a 54th-minute victor with his third league goal in as many matches.
"Expect soon or never", said Mourinho regarding his prospects of signing Sanchez. "So close, so close, so close..." Would it stop me from doing the same again? No.
Juergen Klopp's Liverpool are at bottom club Swansea City on Monday and after their vibrant display against City will be looking to extend a run of four straight wins.
Algerian striker Slimani joined the then Premier League champions in 2016 for a club-record £29m but has failed to provide a spark to the Foxes attack, continuously finding himself behind Jamie Vardy and Shinji Okazaki in the pecking order.
Theo Walcott, fresh from his move from Arsenal, is in line to make his debut for Sam Allardyce's Everton at home to West Bromwich Albion.
Guardiola, who will be without defender Fabian Delph for several weeks after his first half injury, believes City will use their Anfield frustration to bounce back in their next game against Newcastle on Sunday (Jan 21).