Chile forward Sanchez appeared destined to end his three-year stay at the Emirates Stadium by joining Premier League rivals Manchester City in the previous transfer window.
However, Arsenal were not able to capture Thomas Lemar from Monaco, resulting in Sanchez remaining in north London.
Sanchez has found the going tough with just one goal in eight league appearances this term, while sixth-placed Arsenal are already 12 points adrift of swashbuckling leaders City.
Eboue feels Sanchez's quality is not in doubt, though.
The former Gunners defender suggested Arsenal need to retain both the forward's services and those of team-mate Mesut Ozil, whose contract is also up at the end of the season.
"This year is hard. It's hard for him because he wants to move, Arsenal kept him so he has a lot of pressure in his head," he told DAZN.
"Sanchez is a great player. Arsenal have to keep him, Ozil and Sanchez. Anything can happen in football."
November 13, 2004: What a game pic.twitter.com/24DVdmFY7C— Arsenal FC (@Arsenal) November 13, 2017
However, Eboue added that at Arsenal no player is bigger than the collective, and called for everyone to contribute.
"At Arsenal they don't have the two best players. All of them are the best. In the team we don't have the same talent, everybody has their job to do on the pitch," he added.
"Ozil's job is to make assists, the last pass, score goals, sometimes he can shoot, sometimes he can score.
"Sanchez also passes but more goals. In his first year he scored a lot of goals."
At the end of last season, manager Arsene Wenger signed a new two-year deal despite growing discontent from Arsenal supporters at the Frenchman's lengthy tenure.
But Eboue, while understanding the concerns of the fans, believes Wenger should be backed.
"I understand the fans because it's been a long time since Arsenal won the league. I understand the fans but they have to know also that it's not easy.
"It's not easy because now it's not like before where the teams, when they come to play against Arsenal, they were scared before.
"Now when everyone plays against Arsenal, they want to beat Arsenal because when you beat big teams, if you beat Arsenal, if you beat United, City or Chelsea, your week is good.
"It's hard for Arsene Wenger and the fans also. It's very, very hard, but we have to give him a chance.
"Give him a chance and we will see, but I'm always behind him. I'm always behind Arsenal, I'm always behind Wenger, because he took me like a son so I always support him."