Robert Maaskant's men headed into the latest round of fixtures three points adrift of safety, but went ahead through Erik Falkenburg after 32 minutes.
That lead lasted just two minutes, though, as Edouard Duplan restored parity.
Falkenburg struck again four minutes into the second half, before Gevero Markiet made it 2-2.
Kenny van der Weg and Gill Swerts goals gave Breda a match-winning lead, but Rubio Rubin set up a tense finish when he found the net three minutes from time.
NAC held on for a valuable three points that saw them leapfrog Go Ahead Eagles into 16th, goal difference behind 15th-placed Heracles.
Maaskant's side remain in the bottom three, occupying a relegation play-off spot - while Heracles are in the last safe position in the Dutch top flight.
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