The 20-year-old, playing in only his second Tour final, won 5-7 7-6(3) 7-5 and saved four match points before sealing victory.
Third seed Kohlschreiber looked on course for victory when he led by a set and 3-0 but Coric recovered to level the final on a tiebreak. Coric then clawed back a 2-4 deficit in the decider before claiming victory in two hours 38 minutes.
It was a huge week for Coric who has suffered a marked loss of form and has slumped to 79 in the rankings having been tipped to challenge for major honors.
(Reporting by Martyn Herman; Editing by Clare Fallon)