Harrison, 19, took the first set 4-6 from Murray, thanks to mistakes from the Scot and his own impressive ability and movement.
Murray went into the second set still unable to find his rhythm and range of shots, particularly with his forehand. However, he slowly came round and eventually took the set 6-3.
With Murray finding his feet to control the court, Harrison looked as though he had started to tire and Murray took the third set 6-4.
Murray won an early break in the fourth set before seeing it out comfortably 6-2 to progress into the second round of the tournament.