"Like it or not, you need to be patient," he said.
Eakins met with reporters Tuesday, a day after he was fired by general manager Craig MacTavish with the last-place club mired in a deep slump. MacTavish will take over on a transitional basis before handing the reins to interim coach Todd Nelson, who becomes the team's fifth head coach in seven years.
Eakins said he could "speak freely now," but he didn't convey even a hint of bitterness over his half-hour availability. In fact, he sounded like someone who was still with the organization.
The former coach said that, given time, he's confident Nelson and MacTavish will turn things around.
"I understand being out of the playoffs for a long time, but the situation is the situation," Eakins said. "It's kind of like losing all your money. You don't get all your money back the next day.
"You have to start working again and working at it."
The Oilers took a 7-19-5 record into Tuesday night's game at Arizona. Edmonton is on track to miss the playoffs for the ninth consecutive year.
"The fans of this organization: you've suffered greatly," Eakins said. "But I'm going to tell you, you need to be patient. That's the way it goes. You need to be patient and if anybody's going to find a solution, it's going to be Craig MacTavish. I know that for sure.
"His commitment to this organization, his pain that he goes through every day, looking, trying to make this organization better. He will get it done. He will, I know he will."
MacTavish has served as GM for less than two years. Nelson, meanwhile, inherits a team that has dropped 15 of 16 games entering play against the Coyotes.
The Oilers are loaded with young players like Taylor Hall, Jordan Eberle, Ryan Nugent-Hopkins and Nail Yakupov, but a lack of roster depth and experience appears to have hamstrung the squad. Hall, Nugent-Hopkins and Yakupov were all drafted with No. 1 overall picks.
Eakins, who enjoyed AHL coaching success before joining Edmonton in June 2013, guided the Oilers to a 29-44-9 record last season. The team struggled at both ends of the ice again this season and Eakins paid the price.
Nelson spent five seasons at Oklahoma City and guided the American Hockey League affiliate to a 15-6-2-2 record this season.
"I think Nelly's going to bring in a great, fresh approach," Eakins said. "He's an excellent coach. He's fully deserving of this opportunity and he's going to get great support here with the staff."
Eakins also thanked the fans, players and members of the organization for the opportunity, his first NHL coaching job.
"This organization is at where it's at — all of the past means nothing," he said. "Today is where you're at and where do you go forward? There's going to have to be some patience here for Craig to let him do his work."