The goalkeeper says the "table is set" for the next manager to come in and continue the Toffees' success following the Scot's 11-year tenure at Goodison Park
The Scot is on course to lead the Toffees to their seventh straight top-eight finish in the Premier League and Howard believes they can continue that consistency after Moyes departs for the Old Traffor hotseat this summer.
The 34-year-old feels everything is in place for the next manager of the club to come and be just as successful, and the club will not become unstuck.
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"Everyone is saying Manchester United have to have faith in him, give him time to work his magic. Well, it’s the same with the new guy who comes here.
"We are a really strong group of quality players and the foundations are set. The table is set for the next guy and I have no doubt if he’s a strong character, a guy who can lead us well, we will have no problems."
Howard did, however, admit that the atmosphere at the club has been surreal ever since Moyes was announced as Sir Alex Ferguson's successor.
"We’ve not chopped and changed manager in the last 11 years, so it has been surreal, really,” he added.
"Everyone was wanting to know who will be the new manager at Old Trafford and, all of a sudden, it’s your manager.
"It’s a big change, but we have dealt with it. We have come out and played arguably one of our best games of the season against West Ham, spurred on by the moment, the crowd and the occasion.
"Nobody needed to say anything. We have a pretty veteran group of guys, who have been around here and the manager for a while, so it went unsaid, but, as you could see in the performance, it was full steam ahead. I think everyone just knew."