The former Everton manager, who has endured a difficult first season with United, suffered another setback on Sunday with a resounding 3-0 reverse to north-west rivals Liverpool.
And the latest setback has inflicted further damage on the club's hopes of finishing in the top four of the Premier League this season.
Other than a 1-0 victory against Arsenal early in the campaign, United have not beaten any of the top nine sides in the league.
Moyes expected it would be tough emulating the achievements of his fellow Scot, but he has found the task daunting.
"The job was always going to be hard," Moyes said. "Harder? Yes I would say so yes."
The United fans have stuck by Moyes and there were no audible boos at the final whistle on Sunday, but the 50-year-old is aware the team will have to turn the situation around quickly.
"Results will always dictate that but I can only tell you what I've seen today and the supporters were terrific," Moyes added.