As is tradition, the defending men's champion opened proceedings on the show courts on the first Monday of the competition and Murray raced into a one-set lead after only 28 minutes.
Goffin, of Belgium, was also brushed aside 6-4 in the second set but offered slightly more resistance in the third, losing 7-5.
"It was nice," said Murray. "I was nervous this morning and today ... it was nice to walk out to a full crowd for the first match and to a round of applause."
The 27-year-old, who a year ago memorably became Britain's first men's singles champion at the All England Club for 77 years, walked on court to a standing ovation and responded with a clinical victory.
Murray, bumped up to third seed despite slipping to five in the world rankings since his Wimbledon triumph, was immediately in the grasscourt groove, outclassing Goffin in a 28-minute first set that resembled an exhibition match.
The 23-year-old Goffin, who has flat-lined since pushing Roger Federer hard in the fourth round of the 2012 French Open, shook off the early nerves to give Murray a match thereafter.
He even carved out two break points in the fourth game of the third set, only for Murray, with new coach Amelie Mauresmo watching on intently, to respond to the danger.
One flicked backhand down the line by Goffin to hold serve in the next game had the man from Liege smiling, but his resistance snapped at 5-5 when he dropped serve from 40-0 up.
Murray duly held serve in the next game to move through and maintain his record of avoiding first-round defeats at grand slams since an early loss at the 2008 Australian Open.
- Sports & Recreation