The defender helped the Blues overcome Crystal Palace to record a third successive Premier League win on Sunday and then revealed the players know they have to be “on it every game” to retain their place.
Keane played a part in both goals as Marco Silva’s side left it late to secure a 2-0 victory against the Londoners.
The 25-year-old also paired up confidently alongside Kurt Zouma at the heart of defence as he had done in the previous victories against Fulham and Leicester City.
“It’s been good to play alongside him,” said Keane of his link up with Zouma. “We’ve had a bit of consistency now and it’s showing.
“It’s two clean sheets and three wins in a row, so we’ve got to be happy with that and we know there’s strong competition and that we need to be on it every game.
“I feel like that’s pushing us on to do better.
“We’ve got to keep going and see how far that can take us.”
Keane’s battling on the edge of the Palace box allowed forward Richarlison to feed Ademola Lookman for the teasing cross that saw Dominic Calvert-Lewin break the deadlock at Goodison Park three minutes from time.
The defender then drove a long ball upfield just two minutes later from which Cenk Tosun beat goalkeeper Wayne Hennessey to wrap up a win that moved Everton up to eighth in the Premier League table.
However, Keane says it was Jordan Pickford’s heroic 59th-minute penalty save from Palace captain Luka Milivojevic that ultimately swung the game in the hosts’ favour.
“Jordan’s save is massive, it gets the crowd going and gives us a big boost,” he said. “He’s been doing it for a while now. It was top-drawer, a great save, and it was vital to the win.
“You’ve got to keep the belief sometimes. In the second half we were pushing them. We didn’t produce too many clear-cut opportunities but we had a lot of pressure from set plays. We kept it tight at the back and hoped a chance would come and the subs [Lookman and Calvert-Lewin] made a big impact and it was nice to then cap it off with the second.
“I saw Cenk out of the corner of my eye and you hope that if you play those passes hard enough and long enough, the striker will be on to them, and Cenk was. You take your chances and it paid off.
“We’re slowly building a great squad here. You can see now with players coming back to fitness how good we can be. We had a good result today but didn’t play our best and we’ll keep getting better and better.”
Everton’s next outing sees them head to Keane’s first club Manchester United on Sunday.
The Stockport-born Old Trafford graduate added: “We’re above United in the table so we can go there with confidence.
“Obviously, United have turned it around the past few games after a tough patch but we’ve got to go there and play like we have been doing away from home and hopefully we can get a result.”