United confirmed on Sunday they had sold a record 63,000 season tickets for the new campaign.
The figure is viewed as a ringing endorsement of Moyes' appointment as Sir Alex Ferguson's successor.
However, the Scot knows he can only retain that support with performances on the field and, in his programme notes for the Premier League encounter with Chelsea at Old Trafford, he said: "I won't be taking that support for granted.
"I know I have to earn it. I know I have to go out there and show you fans, and everyone at Manchester United, that I'm the right successor."
With no significant additions to his squad heading into the final week of the transfer window, and Wayne Rooney still hoping to be granted a move, Moyes' elevation has not been without it's problems.
However, United's 4-1 win at Swansea last week suggested the players had bought into his appointment, and Moyes is hoping that success continues.
He said: "There is no doubt we are in a transitional period at the club - most people recognise that when a new manager joins - but I'm hoping that it's going to go smoothly and there won't be too many bumps along the way."