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Erwin van de Looi's side went into Friday's Eredivisie clash in fine form, having reeled off three consecutive league victories.
Relegation-threatened Breda made Groningen work hard to make it four wins in a row, but Kostic struck at the death to move his side seven points clear of ninth-placed PEC Zwolle.
Nebojsa Gudelj's team, who are now winless in nine Eredivisie outings and remain just four points above the relegation play-off zone, frustrated their hosts in the first half, but Groningen midfielder Nick van der Velden broke the deadlock from the penalty spot 10 minutes after the interval.
