Bremen took the lead in fortuitous circumstances in the eighth minute when Berkay Dablani deflected Santiago Garcia's low cross into his own net.
Garcia continued to be a threat on the right flank and almost teed up Nils Petersen, but the German forward could not quite get enough on the ball to turn it home.
Elia did double the hosts' lead in the 34th minute when he stroked home from another Garcia cross from the right wing.
Nuremberg, who were second best for the majority of the first half, got a lifeline on the stroke of half time when Hiroshi Kiyotake lashed home a half-volley from the edge of the box after Bremen failed to clear a cross.
The away side were level only for 13 minutes, though, as Elia capitalised on a poor punch by Tim Schafer to rifle a shot into the bottom corner.
However, Kiyotake set up Adam Hlousek to make it 3-3 just four minutes later as Nuremberg refused to be beaten.
Drmic should have given the visitors the lead for the first time on the afternoon in the 73rd minute, but the forward fluffed his effort.
The Swiss international almost redeemed himself with 10 minutes remaining, but his volley on the turn clipped the outside of the post.