Sanchez has cut an increasingly frustrated figure at the Emirates Stadium over the past year and has allowed his contract to run into its final season.
The attacker looked revived somewhat on Saturday, scoring in the Gunners' impressive 2-0 win over north London rivals Tottenham.
And Lauren, a champion with Arsenal's 'Invincibles' side of 2003-04, says Arsene Wenger could taste silverware again if he can manage Sanchez's mood.
"At least we [have to] keep Alexis Sanchez happy," Lauren told Omnisport.
"I don't think he will be the same player as last year, scoring a lot of goals, because if a player said to you at the beginning or the end the season, he wants to leave...
"Either you maintain, keeping him happy, or find issues that make him happy and able to perform the same way as last year, otherwise it will be difficult to maintain the same level.
"The people that are working with Alexis on a daily basis at the club must know about it, but he is a player that, definitely, to be able to win the Premier League, you need to have him in the squad. But you have to make him happy."
After Arsenal's derby win, Wenger spoke of his belief that Sanchez and Mesut Ozil, also approaching the end of his deal, want to remain at the club, but City have been linked with a January raid for the Chile star, while Ozil has also been tipped to move.