The win boosts West Ham's survival hopes but Moyes is still keen to add new players to his squad to help them stay out of trouble.
If Arnautović continues to improve as he has done he could become a great, that is all.
The Austrian was close to unplayable here, at least Huddersfield found him so.
Joe Lolley was hero and villain in the first half, though not in that exact order.
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West Ham were understandably delighted with the result and were equally impressed with the performance from Arnautovic during the game, taking to Twitter to provide their reactions. Noble collects and fires past Lossl.
Huddersfield had the better of a competitive start with attacking trio Rajiv van La Parra, Tom Ince and Lolley leading the charge. Lolley would make up for his embarrassing moment, with a terrific individual finish in the 40th minute to equal the contest.
Antonio Conte's side had drawn 0-0 in two cup games before Saturday's (local time) return to the league and were still unable to force a breakthrough after Leicester went down to 10 men when defender Ben Chilwell was booked twice in the second half.
It was 1-1 into the break, but it surely would not even be that same score even one minute into the second half, as the Hammers took the lead again with Arnautovic scoring superbly off a long kick right at the restart.
Huddersfield were not just overtaken in the table, they were simply outclassed. He was eventually halted close to the penalty spot, but there was so little of the home defence left it was a simple matter for Lanzini to collect the loose ball and belt it into the net.