Anthony Elanga, who was sent off late on, along with United defender Bjorn Hardley and Everton striker Charlie Whitaker following a late melee, scored the only goal of the game after just two minutes to secure a first away win of the league season for Neil Wood's side.
United, who have now won three games on the bounce in all competitions, following on from Tuesday's 3-0 win over Bradford City in the Papa John's Trophy, certainly had to dig in deep for all three points, with Everton looking the more likely to score during the second half.
Nevertheless, the Reds got the job done to secure a second win and first clean sheet of the league campaign, although late drama with three red cards dominated the latter stages.
United came flying out of the blocks, getting their noses in front with just two minutes on the clock. Shola Shoretire, after latching onto Dillon Hoogewerf's pass, chopped inside, wrong-footing Luke Butterfield, and rolled it across the box for Elanga to tap home from close range, burying it through the legs of Everton goalkeeper Jack Barrett.
After weathering an early United storm, the Toffees began to grow into the contest and had their first chance when striker Whitaker, after bringing the ball down well on his chest, volleyed his effort wide of the target by some distance.
Although the Reds always looked threatening on the counter-attack, with Hoogewerf and Elanga often leading the charge, it was Everton who were looking the more likely to score. First, Butterfield brought Matej Kovar into action, forcing him into a routine save from the edge of the box, before Stanley Mills, once again on the volley, connected with Kyle John's cross and forced another save out of Kovar, this time at his near-post.
After scoring early in the first half through Elanga, they very nearly doubled their lead towards the end of the opening 45 minutes through Shoretire. The youngster was picked out by Alvaro Fernandez from the left, brought the ball under his spell inside the penalty area and saw his low drive deflected wide of the target.
United boss Wood was forced into making a change at half time, with Teden Mengi forced off through injury. Player-coach Paul McShane stepped off the bench to replace him.
After scoring early in the first half, Elanga went close to repeating the trick seven minutes into the start of the second 45, this time seeing an excellent free-kick palmed away by Barrett, just as the ball looked set to bulge the net.
Chances inside the opening 20 minutes of the second half were proving to be few and far between. However, the home side did go close to an equaliser when Isaac Price's effort from just inside the box went just wide of the target, after he hit the ball down into the turf.
United were camped inside their own half for much of the second half, but did come close to getting a second when they finally breached the Everton defence. Substitute Joe Hugill got in behind and looked set to put the ball on a plate for fellow substitute Noam Emeran to tap it home, but it lacked power and conviction, before later being denied himself by Barratt from a tight angle.
However, drama came as the clock ticked into the last minute of the 90. Everton won a free-kick and were keen to take it quickly. However, Elanga moved the ball away from Whitaker and it resulted in him throwing the ball at him. A melee then broke out, with almost every player involved. In the end, the referee sent-off United duo Hardley and Elanga, plus Whitaker.
It was harsh, in truth, with nothing more than some pushing and shoving seeming apparent.
In the end, however, United held on to secure another victory.