Austrians Sebastian Proedl and Arnautovic scored midway through the first half as Bremen stormed to a 2-0 lead against a toothless Hoffenheim.
With home fans chanting "We want to see you fight", and jeering them off the pitch at halftime, the hosts looked transformed after the break.
They came out strongly and grabbed a goal in the 50th minute through Sejad Salihovic.
With Bremen keeper Sebastian Mielitz keeping them ahead with a string of superb saves, the visitors dashed any comeback hopes when Arnautovic drilled in a 73rd-minute free-kick.
He notched his fifth goal of the season six minutes later with a nice piece of skill in the box to lift Bremen to eighth on 21 points.
Backed by billionaire Dietmar Hopp, Hoffenheim are third bottom and in the relegation playoff spot after one victory in their last 10 games following their worst start to the Bundesliga.
"This is a bitter defeat because we wanted a lot from this game," Babbel told reporters. "The first half was extremely weak and after the third goal my boys looked a bit broken. Now we are focusing on our game at Hamburg SV on Friday."
Their next opponents drew 1-1 at VfL Wolfsburg with Hamburg leading through Maximilian Beister's first-half goal and the hosts drawing level through a well-timed header by defender Simon Kjaer.
Wolfsburg, who hit the post through Vierinha and could have snatched all three points in the dying seconds when Srdjan Lakic's point-blank header was saved by Rene Adler, moved up to 16 points, a four-point buffer from the relegation zone. Hamburg are ninth on 21.
Leaders Bayern Munich had their advantage cut to eight points over second-placed Bayer Leverkusen after drawing 1-1 with champions Borussia Dortmund in Munich on Saturday.
Leverkusen edged past Nuremberg 1-0 to move up to 30 points with Dortmund on 27 in third.
