United made it seven wins in a row in all competitions for the first time since 2019 as they eased to a 3-1 victory, inspired by another fine performance from Marcus Rashford who scored the third goal from the penalty spot after creating the first two.
Rashford now has five goals in five and seven in as many home games, while this was a 14th win in 17 for Ten Hag's side, who have dragged themselves into the Premier League title conversation during that run while staying alive in three cup competitions.
Friday night was Ten Hag's first experience of the FA Cup and it was a lively one against an Everton side full of fight despite their own league woes.
Conor Coady capitalised on a mistake from David De Gea to cancel out Antony's early opener, only to then score an own goal early in the second half when he met Rashford's cross.
Everton thought they had levelled through substitute Dominic Calvert-Lewin in the 73rd minute but, after VAR intervened with an offside call, Ben Godfrey's foul on Alejandro Garnacho in stoppage time allowed Rashford to finish the job from the penalty spot.
"Everton performed really well at Manchester City (last week) with the draw so you have to be ready for every game," Ten Hag said. "Every player in our squad knows that at 99 per cent you don't win games, you have to be 100 per cent.
"I like the FA Cup, I like the Carabao Cup. It's about do or die. I think we had a team, no, a squad determined to win this game and it was enjoyable again to see the winning attitude is coming and the winning attitude is there in this squad."
The Carabao Cup is up next with Charlton the visitors at Old Trafford on Tuesday night, and Ten Hag may be tempted into changes with Manchester City visiting in the Premier League on Saturday lunchtime. However, he was not giving away his plans just yet.
"I have it in my mind but I've put it in the background," he said of his possible starting XI. "First, I will sleep."
Although Everton's winless run stretched to eight games, under-pressure manager Frank Lampard took encouragement from the fight shown by his players.
And it could have been a different story had Demarai Gray not strayed a fraction offside before playing in the cross that Calvert-Lewin chested home in the 73rd minute.
For all the frustration of Tuesday's capitulation in a 4-1 home loss to Brighton, at Old Trafford the Everton fans stayed to applaud their team in defeat, something Lampard was keen to ensure his players recognised as he led the way in returning the appreciation.
"It was more for the players than for me really, because I think a lot gets said of Goodison the other night and how it became a difficult environment to play in," he said.
"But it is good for the players to understand that when they give everything even in defeat there is a certain respect from the fans and all of them appreciated the players' performance so it was good to stay there for a moment. What we need to do is give them that feeling with results."