Moyes arrived at United's Carrington training complex for his first day of work on Monday, nearly two months after he was confirmed as Sir Alex Ferguson's successor.
"I have worked with Steve, Chris and Jimmy for a number of years and I am delighted they have decided to join me at this great club," Moyes said.
"They bring great qualities in their respective fields and I know that, like me, they feel that this is a challenge to relish. I have great faith that together, we can build upon the success this club has enjoyed over many years.”
Round joined Everton to become Moyes' assistant in 2008 after spending six years as Steve McClaren's number two at Middlesbrough and England.
Lumsden has worked with Moyes since he was manager at Preston North End in the late 90s while Woods has left Everton after 15 years as goalkeeper coach.
United's first-teamers return for pre-season training on Wednesday before Moyes chairs his first press conference on Friday.
Wayne Rooney's Old Trafford future is still yet to be resolved, with Moyes set to be questioned on the situation later this week.