Tottenham consolidated their fourth place in the standings, now just one point behind third-placed European champions Liverpool, which translates into Champions League football next season.
Spurs, 18 points adrift of leaders Chelsea, are six points ahead of fifth-placed north London rivals Arsenal, who drew 0-0 at home against second-placed Manchester United on Tuesday.
Manchester City are ninth, 30 points behind Chelsea after their fourth match without victory.
Defender Sylvain Distin's dreadful error allowed Spurs to go in front just past the half hour with Mido's ninth league goal of the season.
Speedy teenager Aaron Lennon raced into the box on the right and, with Distin looking to let defender Paul Stalteri's ball go into touch, remarkably stole the ball from behind the central defender's back.
Lennon turned the ball inside from the goalline into the path of Mido, who shot left-footed past the outstretched leg of defender Richard Dunne with keeper David James beaten.
With seven minutes remaining Distin lost an aerial challenge outside the box with Mido, and Keane raced on to his fellow striker's header to beat James. He scored with a low shot inside the keeper's right hand post.
Asked if Spurs could finish at least fourth, their England midfielder Michael Carrick said: "You've got to believe...
"It's a long season but we're there at the moment. Hopefully we'll still be there at the end of the season," he told Sky Sports.
Spurs manager Martin Jol said: "I am delighted. To have 40 points in the first week of January is a dream when you start the season.
"We're going to miss Mido," Jol added. The striker is going to play in the African Nations Cup in his home country which starts on January 20.
Spurs may also have lost defender Lee Young-Pyo to serious injury. The South Korean international was carried off midway through the second half after a late challenge by City substitute David Sommeil.
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