Third-seeded Murray, bidding to become the first British man to successfully defend a Wimbledon title since Fred Perry in 1936, hit eight aces and 28 winners. (Agencies)
"I thought it was a very high standard of match, we played some great rallies, and I was glad to finish it in three sets because he was playing very well," Murray said.
"Sometimes you can win in three and not play well but in terms of the way I struck the ball it was a good start,” Murray added.
The British star was never broken as he brushed past the 23-year-old Goffin, the world number 104, with new coach Amelie Mauresmo as well as former NBA superstar Shaquille O'Neal looking on from the Centre Court stands.
Murray, who has made at least the semi-finals on his last five appearances at the All England Club, goes on to face Slovenia's Blaz Rola.
The Scot was joined in the second round by Czech sixth seed Tomas Berdych, the 2010 runner-up, who came from a set down to beat Romania's Victor Hanescu, 6-7 (5/7), 6-1, 6-4, 6-3.
Berdych arrived at Wimbledon having joined US veteran Venus Williams in a naked photo shoot for a pre-tournament ESPN magazine special.
Latvian 12th seed Ernests Gulbis, who made the French Open semi-finals, also went into the second round by defeating Estonia's Jurgen Zopp 7-6 (9/7), 7-5, 7-6 (12/10).
Spanish 18th seed Fernando Verdasco, who took Murray to five sets in the quarter-finals in 2013, was knocked out by Australia's Marinko Matosevic 6-4, 4-6, 6-4, 6-2.
Sergiy Stakhovsky, the Ukrainian who send seven-time champion Roger Federer crashing to a shock second round loss last year, beat Argentina's Carlos Berlocq 6-3, 6-3, 6-3.
Later Monday, Novak Djokovic, the 2011 champion and runner-up to Murray in 2013, starts against 26-year-old Kazakh, Andrey Golubev who is ranked 55 and has lost all three of his main draw appearances at Wimbledon.
Despite being number two in the world, Djokovic is the top seed over current world number one Rafael Nadal.
Former women's world number one Victoria Azarenka won her first match since January on the same court where she suffered a knee injury which ended her 2013 hopes.
Azarenka, the eighth seed, won 6-3, 7-5 against Croatia's Mirjana Lucic-Baroni, who was a semi-finalist back in 1999, on Court One.
That was the same arena where she hurt her right knee in a nasty fall in the first round 12 months ago, an injury which sparked her withdrawal from the next round.
Third-seeded Murray, bidding to become the first British man to successfully defend a Wimbledon title since Fred Perry in 1936, hit eight aces and 28 winners.