The Texas-born golfer looked cool, calm and collected as he gently gave the alligator a nudge by tapping on his tail.
The PGA Tour still orchestrated the draw so that he would be in the featured group in the afternoon with Rory McIlroy and Brandt Snedeker.
Hoffman, a four-time victor on the PGA Tour, carded a six-under 66 to move to 10 under for the tournament and assume sole leadership at Bay Hill, after Grillo and Fitzpatrick had shared it following the first round.
Reigning Masters champion Danny Willett, meanwhile, was a late withdrawal due to illness.
Some notable scores are Justins Rose and Jason Day who are tied for 13th at three-under, with Hideki Matsuyama tied for 20th at two-under par.
At seven-over 151, he was way down in the 120-player field at Bay Hill in a tournament where tributes poured in for "The King", who passed away a year ago.
Starting from the 10th, Fitzpatrick followed six straight pars with birdies on the 16th and 18th to reach the turn in 34, before picking up further shots on the first, third and sixth. A day after Cody Gribble grabbed the tail of an 8-foot alligator and saw it scrambling into the lake, Smylie Kaufman didn't see the alligator.
This year's event is the first since tournament founder Palmer died in September at the age of 87 and numerous tributes are being made to the seven-time major victor.
Palmer died in September, which has created a different atmosphere for this week's tournament that bears his name, the Arnold Palmer Invitational presented by MasterCard.
"I don't play social golf in cold weather", the Australian told Golf Channel.
"It was a tough day and I was glad to get it in at two under", Day said.
In the first staging of the event since the death of Palmer in September past year, his grandson Sam Saunders missed the cut by one stroke.
Emiliano Grillo did his best to stay in the lead with two birdies and two eagles on the day, one of which came on a bunker shot from 88 feet on the 12th hole.
"I felt like it was in there, it was just a matter of trying to put it all together and today I did". "I didn't hit it good off the tee".
Even from there, however, a two-putt was a tall order. I took everything from that day and made sure that whatever I did wrong, I wasn't going to do that again.