The hosts were the form side coming into this clash, so it was no surprise to see them start the strongest.
They piled the pressure on the United back line, and the first chance fell to Beto in the air, but his nod was straight into the gloves of André Onana.
The visitors did break out to swing in a couple of dangerous crosses, but Everton’s relentless pressing eventually proved too much.
United tried but failed to clear a corner, with Abdoulaye Doucouré eventually heading through the melee to Beto, who volleyed off the ground and into the roof of the net past the helpless Onana.
That fifth goal in just four Premier League outings for the Guinea-Bissau international only boosted his side’s momentum, resulting in a doubling of their lead shortly after the half-hour mark.
Beto’s first-time cross was collected by Jack Harrison, whose shot forced a fantastic save out of Onana, but it was ultimately in vain as Doucouré rose highest to meet the rebound and head into the net.
United barely managed an attempt in response to Everton’s first-half dominance, and they were instead lucky that Beto didn’t get his second when Noussair Mazraoui slid in to clear Doucouré’s cross.
Despite slight improvement, United initially still created very little in the attacking third as the second half got underway, and Onana was instead tasked with keeping out Doucouré’s volley.
However, with 20 minutes remaining, Alejandro Garnacho won a free-kick on the edge of the area, allowing Bruno Fernandes to wrong-foot Jordan Pickford and bury into the far corner.
That completely changed the atmosphere on Merseyside, and United pushed to notch the equaliser within 10 minutes.
Another failed clearance from a set-piece played a part, with Manuel Ugarte this time the beneficiary as he volleyed in from the edge of the area.
The equaliser prompted a chaotic finale, and Pickford had to be on top form to tip over Fernandes’ strike to ensure the comeback wasn’t completed.
Then, deep into stoppage time, the Toffees were denied an initially-given penalty by VAR when Ashley Young looked to have been pulled down by Matthijs de Ligt, leading to huge disappointment at the final whistle for the hosts.
They nevertheless remain on an upward trajectory with Moyes at the helm, while United have rarely looked this spirited when going behind this campaign, and their display in the final 20 minutes will surely be cause for optimism for Ruben Amorim.