The 20-time English champions defeated Ajax Amsterdam in the final of the Europa League last week to secure direct entry into the group phase of next season's Champions League, the first time the club has acquired automatic passage to that stage of the competition since the retirement of Alex Ferguson in 2013.
Victory over the Dutch side brought a successful end to a challenging campaign that saw Mourinho's United finish sixth in the Premier League as well as winning the League Cup, and Anderson is confident the Portuguese manager will build on that achievement.
"It's not been pretty to watch this season but, if you look at the stats, at the end of it he's won two trophies in his first season, so whatever he does he wins trophies," Anderson, who was Ferguson's first signing for the club in 1987, told Reuters.
"I think he can improve on the way they play, but I think that will come in time. It's a nice stepping stone by getting to the Champions League and he will adapt to the way Manchester United are expecting to play.
"I think it was a means to an end and he's got his trophies. He's taken some criticism this season from all sides because of the way they play, but I think he will adapt and change his style now they're in the Champions League."
Mourinho's United struggled at times throughout the season, with the club drawing 10 times at home and scoring on only 54 occasions during the league season but, as Anderson points out, the lure of the biggest competition in European club football will allow the Portuguese coach to further strengthen his squad.
"The Champions League is a massive pull for Manchester United," said Anderson, who won the FA Cup in 1990 with United.
"Set aside the money side, it attracts players to the club again and it's a massive advantage for United.
"It's vitally important Man United are competing in these tournaments. The Champions League is a massive tournament and they have to be there. It's in the DNA of the football club and I think Mourinho will build on this now."
(Editing by Amlan Chakraborty)
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