Opening this year's proceedings on Centre Court, Murray started confidently and had the first set wrapped up 6-1 in just half an hour.
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Murray broke early in the second and never looked in trouble on his own serve as he moved further in front by taking the set 6-4.
Goffin was more composed in the third set and had break-point chances on the Murray serve, but couldn't take them and was soon put away.
Murray broke through to go 6-5 in front before finishing the encounter with his eighth ace.
Up next for the Scot in round two is a match against Slovenia's Blaz Rola on Wednesday.