One of the quickest goals in Bundesliga history after just 12 seconds from Adam Szalai gave Mainz a flying start, but Hunt levelled midway through the second half to give Bremen a seven-point cushion over the bottom three.
Mainz edged up to fifth but trail fourth-placed Schalke by three points after failing to build upon their early strike.
Bremen found themselves behind in almost record time with Assani Lukimya losing possession inside his own area to Andreas Ivanschitz, who squared for Szalai to score on the stretch.
Only 12-and-a-half seconds were on the clock at the time of the goal, the quickest in Mainz's history, and the quickest Bremen have ever conceded.
Bremen are no strangers to conceding away from home, though, doing so for the 21st game in a row.
Their response was positive and while they were unable to find an equaliser before half-time, they emerged for the second half with a renewed determination to set the record straight.
They almost conceded a second through Niki Zimling with the post coming to their rescue before Nils Petersen struck the woodwork at the other end.
Schaaf sent Marko Arnautovic on, playing three up front for the final 30 minutes, and the move reaped immediate dividends with the Austrian sending in the cross which Hunt turned past Christian Wetklo in the 69th minute.
Bremen were now in the ascendency with an inspired Hunt seeing another shot tipped past the post by Wetklo.
Arnautovic sent a shot wide with the last kick of the game as Bremen went close to taking all three points, in spite of the early shock.