The Manchester City boss believes the new Red Devils manager has a huge task filling Sir Alex Ferguson's boots
Moyes was appointed by United in May after Ferguson decided to retire following a trophy-laden 27-year spell in charge at Old Trafford.
The Red Devils have made a stuttering start to their defence of the Premier League title, collecting just seven points from their four games so far, leading to questions from some quarters as to whether the ex-Everton manager is the right man for the job.
And City boss Pellegrini feels it is only natural for Moyes to experience greater pressure in this role.
"It is impossible not to be under a lot of pressure replacing Alex Ferguson," the former Malaga and Real Madrid manager stated.
"Managing teams like Manchester United you are always under big pressure - and that's not bad because that is a good way to work because you know what you have to do."
Both Moyes and Pellegrini will get their first taste of the Manchester derby on Sunday and the Chilean is well aware of what the game means to his club and their fans.
"I don't need to be here to understand the importance of the derby against Manchester United, but working here and living everyday here it is more clear that it’s a special game," he added.
Moyes enjoyed a good record with Everton against City during the Roberto Mancini era, losing just one of his eight encounters with the Italian.
However, Pellegrini came out on top when his Villarreal side met Moyes’ Everton in Champions League qualifying in 2005 - a result the Chilean hopes to emulate on Sunday.
"I know he had some wrestles against Manchester City when he managed Everton, but I also had a good record against Everton when he was the manager of Everton!" he quipped.