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.
"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.
