France only needs to win Saturday's doubles match to reach the semifinals.
The 17th-ranked Pouille is France's top player in the absence of No. 10 Jo-Wilfried Tsonga and No. 11 Gael Monfils, while No. 22 Richard Gasquet is also out.
"The atmosphere was great," Pouille said, praising the home support at the Kindarena in Rouen. "I really wanted to start the weekend well and I'm very happy with the way I played."
Edmund, ranked 47th, was filling in for top-ranked Andy Murray, who's nursing an elbow injury.
Edmund was leading 5-2 in the second-set tiebreaker before losing four straight points.
Pouille broke Edmund's serve four times overall, dropping his own twice. Pouille, a quarterfinalist at Wimbledon and the U.S. Open last year, was more decisive in the rallies, hitting 30 winners compared to 20 for Edmund.
"Whether it's on hard courts, on clay or on grass, Kyle doesn't like to move too much, either on his backhand or his forehand," Pouille said. "The objective was to find the right zones. My drop shot worked pretty well today."
Chardy sealed victory by breaking Evans' serve again, clinching the win when Evans could only return his strong forehand pass into the net.
"My tactics were clear in my head and my confidence got better and better as the match went on," Chardy said. "I'm very happy and I hope we go through tomorrow."
In Saturday's doubles, Julien Benneteau and Nicolas Mahut face British pair Dominic Inglot and Jamie Murray.
Britain won the Davis Cup in 2015, beating France in the quarterfinals.
France has not won the competition since 2001, and lost the final to Serbia in 2010 and Switzerland in 2014.