Arnautovic scores hat-trick in win
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Sebastian Prodl gave Bremen the lead after just 21 minutes before Arnautovic registered his first goal in the 29th minute.
Sejad Salihovic pulled a goal back five minutes after the break but Arnautovic confirmed a comfortable away win with two more successful efforts in the last 20 minutes.
Bremen moved into eighth place - two points shy of a Europa League spot - with Hoffenheim remaining in 16th place courtesy of a fourth straight defeat.
Bremen grabbed the lead in the 21st minute from a set-piece as Kevin de Bruyne's free-kick met Prodl, who jumped above Joselu to send a looping header over Hoffenheim goalkeeper Koen Casteels.
Arnautovic doubled the advantage just eight minutes later after Daniel Williams lost the ball deep inside his own territory. Arnautovic played a one-two with De Bruyne before beating Casteels with his left foot.
Bremen had a number of chances during the remainder of the half, including a counter-attack in the 45th minute that saw De Bruyne fire well wide of the target instead of passing to a team-mate.
Hoffenheim pulled a goal back five minutes after the break as Salihovic beat Bremen keeper Sebastian Mielitz to his far corner with a left-footed shot.
The hosts had a number of chances but could not convert any, with Joselu's volley flying just over the crossbar.
Bremen missed a couple of chances in the 72nd minute as both Lukas Schmitz and Theo Gebre Selassie came close.
But Arnautovic re-established a two-goal lead a minute later with a curling free-kick that flew over the wall and dipped into the top-left corner.
And the Austrian completed his hat-trick in the 79th minute, running onto a through ball from Joseph Akpala and chipping it past Casteels.