Gallacher is one of a clutch of players on one under par at Augusta. Fellow Scots Sandy Lyle and Bradley Neil are two over and six over, respectively.
"Any time you shoot under par round here, it's a good score," Gallacher told BBC Sport.
"I plotted my way round, holed a few putts and didn't make many mistakes."
Justin Rose holds a share of the clubhouse lead on five under.
After bogeying the first hole, Gallacher birdied the fourth, 10th and 15th - describing the latter as "not orthodox" - before bogeying the 16th.
"You've just got to try and accept where you hit it and take the next shot on," he added.
"You can't get over aggressive. It's a week of patience and just trying to build on it. You're one score away from being right up there."
"Its really, really hot out there and quite a hilly track, so it takes it of you mentally and physically."
Amateur champion Neil picked up 10 bogeys and four birdies, while 1988 winner Lyle bogeyed four holes and birdied holes two and 12.
Scotland's Stephen Gallacher highlighted a need for "patience" after opening the Masters with an opening-round score of 71.