The Brits were upset by a Grigor Dimitrov-inspired Bulgaria on Friday night, Jamie Murray and Joe Salisbury losing the crucial doubles match in a sudden-death champions' tie-breaker.
With only the group winners guaranteed a place in the quarter-finals, the Brits knew they almost certainly had to beat Belgium to retain any chance of progressing.
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Their efforts endured the worst possible start on Sunday night as Steve Darcis beat Cam Norrie for the loss of just six games to leave Evans needing to beat Goffin, ranked 31 places above him in the world.
But the Brit produced a spectacular display of clean hitting to dispatch his opponent in one hour and 50 minutes, one of the best wins of his career and only the sixth time he has beaten a top-20 player.
Great Britain, who had to call up James Ward to replace former world No 1 Andy Murray after he pulled out of the Australian swing with a pelvis injury, now need to beat Moldova on Tuesday to reach the quarter-finals.
In Group C's other tie, Dimitrov continued his good form to help Bulgaria to victory over Moldova and top of the standings.
Elsewhere, Australia beat Canada 3-0 despite losing Nick Kyrgios to a troublesome back while Italy beat Norway via a deciding doubles rubber.
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