Wes Brown's second-half own goal was enough for the Blues, and the home side's efforts to avoid relegation now appear doomed after this afternoon's reverse.
The Blues spurned a couple of good openings inside the first 15 minutes when Steven Naismith hit a shot high and wide in a good position, after Gerard Deulofeu's shot was saved after a strong run took him inside the Sunderland area.
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The game sprung into life on the hour mark with a glut of chances for both sides.
Naismith then missed the best chance of them all when he clipped wide of an open goal after Vito Mannone had got himself into trouble with a poor clearance.
The end-to-end stuff continued and Johnson had a half-chance after Ross Barkley had provided the Blues with some addead impetus from the bench.
With the game being played at a frenetic pace, it was eventually put in inadvertently by Brown when the ball hit him on the hip and left Mannone stranded with 15 minutes to play.
Ki wasted a great a chance with a tame side-footed effort after Johnson had played his teammate in. Howard fumbled the shot, but the Toffees managed to scramble it clear.