Belgium had lost their last four meetings with France, including in the 2017 final. They will face either No.1-ranked Italy or Austria on Friday. The last two quarter-finals will be held on Thursday, including No.2 Germany against Argentina and Spain against the No.4-seeded Czech Republic.
Bergs was in control throughout on Tuesday and sealed the match with a superb forehand winner down the line.
Earlier, Raphael Collignon rallied past Corentin Moutet 2-6 7-5 7-5 to give Belgium a 1-0 lead. Ties are played as a best-of-three matches: two singles and one doubles.
Both nations were back at this stage of the team competition since it was revamped in 2019. The Davis Cup Final 8 in Bologna is the sixth edition of the event to have the champion decided at a neutral site.
Moutet played solid tennis until the 12th game of the second set when he hit two double faults and fluffed an easy point at the net after trying a between-the-legs shot, dropping his serve as Collignon forced a decider.
A similar scenario unfolded in the third set as Collignon made the decisive break in the final game.
"It was a crazy match," Collignon said.
"I think I was maybe a bit lucky that he chose to do this tweener at 6-5 because it was tough for me. I was not playing so good. He was dominating the match."
Italy are without four-time grand slam champion Jannik Sinner, last week's ATP Finals winner in Turin.
Spain will be missing Carlos Alcaraz, the man Sinner beat in the decider, who withdrew earlier on Tuesday because of a hamstring injury.
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