Stuttgart were leading 2-1 before an extraordinary finale in which their forward Martin Harnik was sent off for a second bookable offence and Jannik Vestergaard headed Bremen level with four minutes left.
But against all the odds, Serey Die split open the Bremen defence with a through ball to Ginczek who kept his cool to score the winner and throw his team a lifeline in the battle against relegation.
Stuttgart climbed to 17th in the 18-team table with 26 points, one behind Paderborn, who are in the relegation playoff place, and three adrift of Hanover 96 and safety.
Their win left Hamburg SV, the only ever-present team in the Bundesliga's 52-year history, bottom of the table with 25 points and in ever increasing danger of losing their proud record.
Christian Gentner volleyed Stuttgart in front after 15 minutes and they kept their lead until five minutes into the second half when Davie Selke headed Bremen level.
Harnik, who had a topsy-turvy afternoon, missed two clearcut chances in quick succession but then atoned when he took advantage of a misjudged foray by Bremen goalkeeper Raphael Wolf to set up Ginczek, who headed into the empty goal.
Cologne beat Hoffenheim 3-2 in Sunday's other game, a meeting of two mid-table teams.
(Writing by Brian Homewood in Berne, editing by Ed Osmond)
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