Needing to beat Belgium to stay alive following their opening-day Group C defeat by Bulgaria, Britain got off to the worst possible start when Cameron Norrie suffered a surprise straight-sets defeat to veteran Steve Darcis.
However, British No 1 Evans defeated world No 11 Goffin 6-4, 6-4 to send the tie into a deciding rubber and doubles pair Murray and Salisbury duly obliged with a 6-3, 7-6 (7) win over Sander Gille and Joran Vliegen.
Tim Henman’s team now need to beat Moldova in their final round-robin match tomorrow to reach the quarter-finals.
Evans, who let slip a one-set lead against Bulgaria’s former world No 3 Grigor Dimitrov on Friday, held the upper hand throughout most of his first ever meeting with Goffin despite winning only 57 per cent of his service points.
He secured all three of his break points in the first set but was broken twice himself as both men struggled to hold serve.
The 29-year-old Briton eventually served out the first set and broke his opponent twice early in the second to race into a 4-0 lead. However, Goffin then reeled off three games in a row to stop the rot and set up a nerve-jangling finish.
Evans squandered four match points and was forced to save a break point before eventually prevailing after one hour and 48 minutes to secure a first win over a top-20 player since eliminating Nikoloz Basilashvili at Wimbledon last year.
“I felt the level was good out there, I’d say one of the highest levels I’ve played,” Evans told the ATP Cup website. “It felt that way and obviously it made it pretty sweet to have everyone there behind.
“And in this new format, I think it’s a pretty good way to play having the seats behind in the corners. It’s pretty cool. It definitely helped to get me through the match.”
Murray and Salisbury lost their deciding match against Dimitrov and Alexandar Lazarov on Friday but made amends at Ken Rosewall Arena with a battling victory in which they denied their opponents a single break point.
The Britons broke in the fourth game and served out the first set 6-3 but there was little to choose between the pairs in the second set.
Gille and Vliegen led 3-0 and 5-3 in the tiebreak before Murray and Salisbury twice levelled and the Britons squandered two match points before finally taking the breaker 9-7. “Evo played an amazing match today,” Murray said of Evan’s performance. “Me, personally, I don’t know how Joe felt, but I was, like: ‘I cannot let that slide’.
“We’ve got to win that match and make his performance count for something.”
Earlier, Darcis needed only one hour and 12 minutes to beat world No 53 Norrie 6-2, 6-4.
The 35-year-old Belgian has announced his intention to retire after the Australian Open but earned his best win since beating Adrian Mannarino at an ATP Challenger Tour event in Lisbon last May.
Bulgaria earlier clinched their place in the last eight by defeating Moldova 2-1, Australia beat Canada 3-0, despite losing Nick Kyrgios to a troublesome back, and Karen Khachanov and Daniil Medvedev helped Russia condemn the United States to a second successive defeat.