Argentine Di Santo converted his spot-kick for his 13th goal of the season after Hamburg's Valon Behrami had brought down Zlatko Junuzovic and was sent off as new Hamburg coach Bruno Labbadia made a losing return to their bench.
Werder moved up to seventh on 38 points with victory in the northern derby, leaving Hamburg in last place on 25 with five matches left in the season.
Labbadia returned to Hamburg this week, tasked with saving their proud Bundesliga record. Hamburg are the only team never to have played in the second division since the creation of the Bundesliga in 1963.
Second-placed VfL Wolfsburg, on 60 points, will look to cut Bayern Munich's lead back to 10 points when they take on Schalke 04 later on Sunday.
The Bavarians are top after their 2-0 victory over Hoffenheim on Saturday. Bayer Leverkusen are third on 54 points following their 4-0 demolition of Hanover 96.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
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