After Raphael Collignon battled back to beat Corentin Moutet, world number 43 Zizou Bergs then took to court and beat Arthur Rinderknech 6-3, 7-6(4).
Bergs was in commanding form but wobbled when serving for the match at 5-4 in the second set.
Rinderknech had a set point to level the match but could not take it, and Bergs then regained control to win the tiebreaker, thumping a massive forehand winner before sprinting towards his teammates for a joyous celebration.
Moutet had looked in control against Collignon when he took the first set 6-2, but then fell apart at crucial moments.
At 5-6 in the second set, Moutet twice double-faulted and also played an inexplicable trick shot that backfired on a crucial point.
Collignon played rock-solid tennis and Moutet crumbled again at 5-6 in the decider, double-faulting on match point down.
Belgium will face holders and hosts Italy in the semi-finals later this week if the Italians get past Austria on Wednesday.
Italy are without four-time Grand Slam champion Jannik Sinner while Spain, who face Czech Republic on Thursday, are deprived of world number one Carlos Alcaraz after he withdrew with a muscular injury on Tuesday.