The goal was scored in the 51st minute after a flowing move that started with left-back Adam Drury and progressed via Darryn Huckerby and Paul McVeigh who set up Earnshaw.
Sunderland threw everything at Norwich after going behind, but found Paul Gallacher and the City defence in determined mood. Even five minutes of stoppage time failed to produce an equaliser for the Black Cats.
The visitors kicked off attacking the River End goal and in the second minute Gallacher made a very good save to keep out David Connolly's low shot.
Huckerby then went on a good run that ended with a shot that was deflected for a corner.
Tobias Hysen's free-kick skimmed over the Norwich crossbar as both sides continued to produce some fluent passing moves, before Hysen brought a save from Gallacher with a low shot on the half-hour.
Stephen Caldwell was booked for clattering into Huckerby two minutes later.
Just six minutes into the second half, a flowing move down the Norwich left saw Huckerby feed McVeigh and his pass released the lively Earnshaw who beat Darren Ward low to his right with a crisp right-foot shot from 18 yards.
Graham Kavanagh was then booked for throwing the ball at Huckerby as the City winger was regaining his feet after being fouled.
Daryl Murphy replaced Hysen after 67 minutes and Norwich brought on Andy Hughes for McVeigh as the visitors stepped up their pressure on the home defence.
In a last throw of the dice Keane brought on Stephen Elliott for Connolly while, for Peter Grant's side, Jurgen Colin limped off after 84 minutes to be replaced by Robert Eagle.
Although they huffed and puffed the Black Cats could not break down the City rearguard in which veteran Dion Dublin was outstanding.
The result leaves the Canaries 13th in the Championship table while Sunderland have three fewer points and are 17th.