The 20-time English champions dominated the contest at Old Trafford but appeared to be heading for a frustrating goalless draw, with Omonia goalkeeper Francis Uzoho putting in an outstanding performance.
Man United secured all the points in second-half stoppage time, though, with Scott McTominay coming off the bench to fire the winner at Old Trafford.
Fellow substitutes Christian Eriksen and Jadon Sancho were also involved in the build-up, while the likes of Anthony Martial, Marcus Rashford and Cristiano Ronaldo have come off the bench to influence recent matches.
Ten Hag has said that Man United's success this season will be determined by their squad, with the Dutchman pleased that players off the bench are having positive impacts.
"We are happy with that and that tells you something about the spirit in the team and the squad," Ten Hag told reporters after the clash with Omonia. "That they understand it is not about 11 players, it's about the squad.
"You have to do it and sometimes it works like that. Of course, when you believe in it, and an opponent is getting tired, then subs can have an impact, when they have the right attitude, when they have the right focus to be direct.
"In offence, but I also like defence. For example, Raphael Varane, he came in and he had a massive defence action [at Everton, when he made a saving header].
"When you bring in Scott McTominay, you know, of course, that he can arrive there. But it is not your typical scorer who will score you 20 goals, that's clear when you bring on Marcus Rashford.
"We create an energy and they bring a tempo over the left wide area and that is good to see. And it can be that your sub will score. But when I say also, players have other tasks like Rapha Varane or Scott McTominay, like to save the win."
Ten Hag decided against making wholesale changes to his starting side against Omonia on Thursday night, despite the team welcoming Newcastle United in the Premier League on Sunday afternoon.
Fred, Tyrell Malacia and Ronaldo were the three players to come in from the Everton clash, with Luke Shaw, Eriksen and Martial dropping out.
Varane was again on the bench following an ankle injury, but the Frenchman is likely to return to the starting side against Newcastle this weekend.
Man United have now won eight of their last 10 matches in all competitions, including their last three, with the team preparing to play five more games before the end of October.