Garner's smart finish put the Toffees ahead at the Hill Dickinson Stadium before Hackney's volley late in the first half left United with a mountain to climb after the break.
Magpies head coach Matthias Jaissle made one change to his side for the trip to Merseyside, with Jacob Ramsey coming in for Joelinton in midfield.
Kiernan Dewsbury-Hall saw his effort deflected wide in the first attack of the game for the hosts, while Lewis Hall sliced a shot high and wide from distance at the other end.
Looking to follow up Sunday's Premier League triumph at Crystal Palace with another win, United started confidently, with Jordan Pickford having to parry Harvey Barnes' header after Nico González picked him out with a raking crossfield ball. Jacob Murphy then saw his effort blocked before Pickford spilled Joe Willock's attempt from a tight angle.
In comparison to the Toffees' England stopper, Lukáš Horníček enjoyed a quiet start to the game, but was called into action by Hayden Hackney's tame half-volley that the Czech international gathered with ease. He might have had to react swiftly to repel Tyrique George's thumping effort from the edge of the box, had Dan Burn not thrown himself in the way of it.
United frontman Will Osula was denied by a last-ditch James Tarkowski challenge just before the half hour but five minutes later, Everton moved in front. Pickford's long ball was chased by George, who robbed Amar Dedić on the right. Garner picked up possession and he whipped a fine first-time effort past Horníček and into the bottom right-hand corner.
Just before the break, Barnes lashed wide before tumbling under the challenge of Dewsbury-Hall in the box, but referee Farai Hallam was unmoved by the visitors' appeals for a penalty. Just moments later, they fell behind after Hallam played an advantage after Ramsey's foul on Dewsbury-Hall, allowing Thierno Barry to pick out Hackney in the box. The former Middlesbrough midfielder's finish, low and arrowed into the corner, was unerring.
**Newcastle United:** Lukáš Horníček, Lewis Hall, Fabian Schär, Nico González, Will Osula, Harvey Barnes, Jacob Murphy, Joe Willock, Dan Burn (c), Amar Dedić, Jacob Ramsey
**Substitutes:** Nick Pope, Sven Botman, Joelinton, Aladji Bamba, Yoane Wissa, Malick Thiaw, Sean Steur, Bazoumana Touré, Anthony Elanga