The pressure on Moyes has ratcheted up following the 2-1 FA Cup defeat by Swansea at Old Trafford, the fourth home defeat in a month for United.
Brown, who played a role in Moyes' development as a coach by allowing him to watch Scotland's training sessions at the 1998 World Cup, believes the current United manager will have a change in fortune and points out Arsenal were viewed as being in crisis at the start of the season.
It took almost four years for Sir Alex Ferguson to win a trophy after he took over as United manager in 1986, and Brown said that example should also be borne in mind.
He said: "I don't think anyone should be ringing the alarm bells.
"The best example is Arsenal at the start of the season, all the talk was about Arsene Wenger and the club being in crisis and look at them now - top of the league.
"All it needs is a couple of months of good results and all this will be forgotten and David Moyes will do that - there is no doubt in my mind.
"Alex [Ferguson] didn't go into Manchester United and wave a magic wand and make it okay and although David is in a slightly different situation it is similar in some ways.
"Replacing the best in the business was always going to be difficult but David Moyes is a strong character and an intelligent guy and I know he is respected by the players.
"People can be disappointed by recent results but they should accept that he is an accomplished manager and be certain the team can recover."
Neil Warnock, however, believes Moyes' job is being made more difficult by Ferguson now sitting in the directors' box watching over him.
The widely-travelled manager, whose former clubs include QPR, Sheffield United and Leeds, said: "You've got Sir Alex Ferguson sitting in the stands and I don't think that helps."