The Serbian midfielder curled home a low drive in the 87th minute. Julian Baumgartlinger's headed own goal had put Hamburg ahead in the first half before Yunus Malli levelled for the hosts in the 76th minute.
Victory lifted Hamburg, the only team to have played every season in the Bundesliga since its creation in 1963, up to 14th on 31 points, one point above the relegation playoff spot with three games left in the season.
Mainz, who had defender Daniel Brosinski sent off in the 89th minute, lost Colombian Elkin Soto to a serious knee injury after half an hour.
He was stretchered off after a clumsy challenge from behind by Rafael van der Vaart.
The defeat was a bitter blow to Mainz as it all but ended their hopes of playing European football next season and left them 10th in the table on 37 points.
Bayern Munich, who have already clinched the title, fielded a second string team in a 2-0 defeat to Bayer Leverkusen on Saturday as they prepare for their Champions League semi-final against Barcelona next week.
Borussia Moenchengladbach can reclaim third place from Leverkusen when they take on Hertha Berlin later on Sunday.
(Reporting by Karolos Grohmann; editing by Toby Davis)
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