They threatened when Pedro sent an acrobatic bicycle kick off target and, after Idrissa Gueye was booked for a late challenge on Cesc Fabregas, had another chance when Willian curled an effort wide of Jordan Pickford's post.
Everton midfielder Gylfi Sigurdsson tried his luck but saw a sliced strike sail off target -- but with five minutes remaining until the break, the visitors were two down as Morata headed Cesar Azpilicueta's perfect cross beyond Pickford.
Chelsea began the second half brightly, Pedro firing in another effort, before Everton made and missed a chance to get back into the match when Rooney played in Sandro only for the forward to scuff his shot.
They again came close to finding a third goal on the hour mark when Azpilicueta's cross caused havoc but no home player was able to get the final touch, and soon afterwards Moses strode forward again but sent his low shot straight at Pickford.
And although defender Ashley Williams shot over as Everton mustered some late pressure, with Thibaut Courtois also tipping a Gueye effort over, but Ronald Koeman's side were unable to shake Chelsea out of their comfort zone.
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