Norwich did pressure the visitors on occasion in the first half, but it took until stoppage time for Lukas Rupp to test Everton keeper Jordan Pickford with a shot on target.
Everton upped the tempo considerably in the second half, taking the lead in the 55th minute when Keane sent a glancing header from a corner flying past Norwich keeper Tim Krul into the net.
Rather than sparking the home side into life, Norwich looked completely rudderless after they went behind, and the closest they came to an equaliser was when substitute Adam Idah forced a late corner.
Norwich remain bottom on 21 points, six points behind 17th-placed West Ham United with seven games left to play, and manager Daniel Farke said he can still see a way for his side to avoid the drop.
“I can just praise the workload, the togetherness. In terms of attitude, we showed today organisation and I can just give compliments,” he said.
“When you wear this yellow shirt you have responsibility and big pride. As long as there is a chance, we won’t give up.”
Everton, whose boss Carlo Ancelotti said they were aiming for European football next season, are 10th on 41 points, four behind seventh-placed Tottenham Hotspur.
“What we needed from this game was three points, the three points arrived and now we can dream to climb up the table. The players satisfied me a lot,” Ancelotti said. (Reporting by Philip O’Connor Editing by Toby Davis)