The visitors started much the better of the two teams and were unfortunate not to edge early in the contest.
Debutant Havard Nielsen might have marked his first game for Eintracht in the style after just eight minutes, only for a last-ditch block to deflect the his goal-bound effort over the crossbar.
Bremen grew into the encounter and were unlucky not to open the scoring early in the second half.
Robin Dutt's men thought that they had taken the lead on the hour mark when Eljero Elia got on the end of Nils Petersen's flick-on and fired beyond Daniel Davari, but his effort was chalked off for offside.
It proved as close as the hosts would come to breaking the deadlock as the visitors held out for a point in their fight against relegation.