David Wagner saw his team's good start to the contest go away in the 25th minute, when misplays by Jonas Lossl and Joe Lolley lead to Mark Noble taking advantage to score his 50th career goal for a 1-0 lead.
For the third successive away game, Arnautovic found the back of the net for the Hammers, his latest strike coming 11 seconds into the second half at the John Smith's Stadium.
The Austria worldwide was instrumental in the game and created the next two goals, both finished by Manuel Lanzini to give West Ham the three points.
Lolley made amends in great style five minutes before the break, beating West Ham keeper Adrian with a fine curling left-foot shot to send the teams in level at the interval.
However, it was West Ham who broke the deadlock just before the half hour after a mistake by Lolley from Lossl's short goal kick.
West Ham United completed the season sweep of Huddersfield and unfortunately won't play them again, at least in the Premier League, for the rest of the season. Alex is a young British player who arrives as one of the top performers in the Sky Bet Championship and with a desire to prove himself in the Premier League.
The second half couldn't have started any worse for Huddersfield.
A long ball from Angelo Ogbonna was flicked onto the Austrian by Cheikhou Kouyate, he lifted it over Tommy Smith and lashed beyond Lossl to restore West Ham's lead.
West Ham did make it 3-1 soon after as Lanzini raced clear and fired across goal into the bottom corner.
"The board know - and they want to add a few players - it's just the difficulty of the month", Moyes continued.