Fish, who was playing for the first time in 18 months after spending a spell on the sidelines following struggles with a heart problem and an anxiety disorder, looked rusty in the first set and appeared set for a quick exit.
However, the 33-year-old veteran clawed his way back into the match by taking the second set 6-4. Harrison would not be quelled though, taking the third set tie-breaker to seal the match.
It was Fish who stole the spotlight though for an incredible gravity-defying moment when he flipped over the net. However, before he could fall over to the other side, Harrison ran over and helped him back to an upright position.
Speaking about his comeback, Fish said: "It felt fantastic to be out there. I've worked really hard to get back in shape so I don't have any issues during a match or after a match.
"I've worked extremely hard to put myself in the best position to not worry about things while I'm out there because if I was out of shape or if I didn't feel well or if it was going to be a long, hot match, a lot of things creep into your head.
"That's one of the things I need to do on the court is to be pretty even keeled and positive. I used to be very hard on myself at times, a lot of times, really. It's something I have to do. If I don't then I can get into trouble. It's something I've worked on with my sports psychologist, what kind of mind frame you need out there."
Fish's anxiety problems began during the 2012 US Open, where he withdrew from the tournament with an irregular heartbeat when he was supposed to face Roger Federer in the fourth round.
After undergoing a minor surgical procedure to correct a heart arrhythmia, Fish started suffering from frequent panic attacks. The American has now developed techniques to manage his anxiety and says he is determined to keep his tennis career alive.