With Leipzig dominating the game and leading after a Lois Openda goal, Njinmah shed two defenders and blasted a long-range effort into the net from outside the box with 15 minutes remaining.
Bremen, promoted in 2022 after a year in the second division, have now opened up a six-point gap on 15th-placed Union Berlin in what is likely to be a season-long relegation scrap.
Leipzig bid farewell to long-time servant Emil Forsberg, who played his last match for the club before moving to New York Red Bulls after nine years in Saxony. Leipzig striker Yussuf Poulsen looked to have headed the visitors in front just before half-time, but a VAR review showed the Denmark forward was offside in the build-up.