The Sunderland boss takes his side to Old Trafford on Wednesday and the Uruguayan knows his opponents are in desperate need of a win
United have struggled under David Moyes' stewardship following Sir Alex Ferguson's retirement last season, with a worst league start in 24 years followed by their worst home form in over a decade.
And Poyet sympathises with Moyes, noting that the transitional period was always going to be a tough task, and feels United need to reach the Capital One Cup final more than his relegation-threatened Sunderland.
"David took the most difficult job in the world after Sir Alex left," Poyet told reporters. "It was always going to be the most difficult job in the world.
"It's a big change, and it's not like when you change managers at any other club, after two or three years. I hope he's given time to change things round because he deserves that for what he's done at Everton."
United head to Old Trafford trailing 2-1 from the first leg following an own goal by Ryan Giggs and a Fabio Borini penalty, with the winners facing a Manchester City side fresh from a record 9-0 aggregate win over West Ham.
Poyet said: "If somebody needs to win, then of course it's Manchester United. Everybody in the world will be expecting them to win, but because of that, there's more pressure on them.
"It could be a must-win for them because it’s a possibility to win a trophy, it’s a possibility of beating their biggest rivals Manchester City to win a trophy. It’s an important game for Manchester United, for sure.
"We're not going to go there to feel sorry for them, we are going to go there and do our job.
"In normal circumstances, If Manchester United were in the top four or top two and we were mid-table, it would probably be a bigger game for us, it would be massive and a unique opportunity.
"But now, because of where we are in the league and what's happening in the league, it makes it quite a strange game. Both teams have a chance to win and change the course of their season."