The Bundesliga's only ever-present side are at risk of losing their top-flight status after a miserable campaign and Mirko Slomka's men failed to settle any nerves in a frustrating first leg at the Imtech Arena.
The hosts created good openings in the first half but their two main threats, Pierre-Michel Lasogga and Hakan Calhanoglu, both failed to stretch goalkeeper Wolfgang Hesl.
It was a similar story at the other end as Greuther, who finished third in the second division this season, wasted chances through Zoltan Stieber and Ilir Azemi.
The visitors should have taken the lead early in the second half when Niko Giesselmann found space at a corner, but the unmarked defender headed wide.
Calhanoglu headed tamely on target soon after, with Hesl making a comfortable save.
Hesl faced one more modest test in the closing minutes as he saved Lasogga's low shot with his legs.
The draw does little for either side, although Hamburg will be grateful not to have conceded an away goal going into the second leg on Sunday.
