Strachan received the backing of the SFA board last week despite his side sitting fifth place in their World Cup qualifying group.
Speculation had linked Sunderland manager Moyes with the job.
"I don't think there's anybody out there who could do a better job than Gordon Strachan," said Moyes, whose side beat Hull City 3-0 on Saturday.
The Scot had been under pressure in the first year of his own four-year contract with the English Premier League club.
But the win over Hull lifts Sunderland above Swansea City and off the bottom of the table.
Moyes has often said that he would like to manage his country at some point but was unhappy to be linked with the role with Strachan still in post.
"Quite rightly so that they kept Gordon on, because he's got a limited group to choose from," said the former Everton, Manchester United and Real Sociedad boss.
"I think Scotland did well against England. I think that the score flattered England a little bit.
"And I think Gordon's working with the tools he's got and deserves to be in the position."
Scotland's 3-0 defeat by the English at Wembley left the Scots six points behind the Group F leaders and four adrift of second-top Slovenia.
However, Ross County winger Chris Burke, who has seven Scotland caps, thinks Strachan's side can still finish second and secure a play-off place.
"The way we have to look at it is - are we going to finish first, or did we think we were going to finish first?" he told BBC Scotland.
"No, we weren't. Realistically, we were looking at coming second and is that still a possibility? Of course it is.
"It is only four points and the team that are in second are coming next to Hampden and, if we beat them, it is back on.
"One thing about Gordon is he is a fantastic coach and I've learned so much in my game when I was at the age of 30 when he introduced me back into the Scotland squad."
Former Scotland left-back Gary Naysmith agreed about Strachan's qualities.
"Everybody says on the training pitch, you won't find anyone better," said the East Fife manager.
"I've never heard one person say anything bad about Gordon Strachan's coaching.
"I thought there was a lot of optimism from the England game, but I can also understand when you look at the statistics why some of the fans are wanting a change.
"But you hear what David Moyes had to say there - another manager would have the same group of players.
"By Gordon staying, he obviously believes we can still qualify, because I firmly believe that, if he didn't think they could qualify or get into second place, he would have walked out the door."
David Moyes thinks the Scottish FA did the right thing by retaining Gordon Strachan as national head coach.