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The Reds regained their 12-point lead in the Premier League title race but were made to sweat after multiple Everton scares in the first half.
Beto had the ball in the back of the Liverpool net in the 20th minute before his goal was ruled out for a narrow offside.
It was the 100th time that Liverpool have beaten Everton and they now need a maximum of 13 points from their final eight league games to clinch the title.
Moyes was hoping for his first ever win at Anfield at the 22nd attempt.
The Everton centre-back and hero of Feburary's derby thundered into a slide tackle on Alexis Mac Allister, booming the ball away but catching the Liverpool midfielder high on the leg with his studs.
On-field referee Samuel Barrott gave Tarkowski a yellow card but after replays, it felt like VAR was bound to intervene.
Everton legend Duncan Ferguson went as far as to say that Tarkowski 'knew what he was doing', admitting that it should've been a 'straight red'.
The Premier League explained the decision by posting on their Match Centre account: "The referee's call of yellow card for a reckless foul by Tarkowski was checked by the VAR, with contact on the follow through after Tarkowski had played the ball deemed to be reckless."
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James Tarkowski was only given a yellow card for this tackle(Image: Liverpool FC/Liverpool FC via Getty Images)
Throughout all of David Moyes' 21 previous trips to Anfield - none of which ended in a win - he'll have dreamed of the kind of chance which fell to Beto midway through the first way.
Jota created the chance by pick-pocketing James Garner before Luis Diaz flicked the ball back into his path.
Jota then took it away from both Idrissa Gana Gueye and Tarkowski, finishing past Jordan Pickford with aplomb.
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