Goffin beat Kyle Edmund in the first Davis Cup final match but Murray’s victory over Ruben Bemelmans levelled matters.
The two-time Grand Slam champion partnered brother Jamie in the doubles victory over Goffin and Steve Darcis on Saturday, meaning Murray went into Sunday’s clash with the Belgian number one knowing a win would seal the title for Britain for the first time since 1936.
Murray had won both of his previous meetings with Goffin, but had never faced the 24-year-old on clay.
The Scot started strongly and broke in the sixth game of the first set, having survived a break point himself in the previous game, to take the advantage.
Three set points came and went for Murray in the eighth game but he held serve in the ninth to take the opener.
Goffin fought back in the second set and forced a break point in the second game but Murray survived before breaking himself to take a 6-5 lead.
The 28-year-old held his nerve to take the second set 7-5.
Goffin responded well and he finally managed to force a break in the third set to take a 2-0 lead but Murray hit back immediately with a break of his own.
Murray broke again to take a 4-3 lead before holding serve to move 5-3 ahead.
The former US Open and Wimbledon champion could sense the victory and moved 40-15 ahead to set up match point. Goffin survived the first but he could do nothing about the second as Murray produced an exquisite lob as the Belgian had moved to the net to win the match and the trophy for Great Britain.
Speaking to the BBC afterwards, Murray said: “I would imagine everyone will be on fire tonight, it’s been an unbelievable road to get here and we’ve got to enjoy this because we might not get another opportunity to do this again as a team and we should celebrate tonight.”