The Austrian fifth seed saw off the big-serving German 7-5, 6-3 after Greece's Tsitsipas, making his debut at the tournament, had earlier beaten Federer 6-3, 6-4.
The Spanish top seed, who has never won the season-ending event, came into the tournament at London's O2 Arena under an injury cloud and was well short of his imperious best.
Zverev won the season-ending tournament last November, beating Roger Federer and Novak Djokovic back-to-back.
Medvedev made a fast start, winning his service game and then breaking Zverev's for an early 2-0 lead.
Zverev, ranked sixth in the world but suffering a poor season, was pushed all the way by the 71st-ranked Chardy before finally emerging in one hour and 48 minutes.
Zverev fired a cross-court forehand winner to clinch a 6-4, 3-6, 10-4 victory in the decisive singles match and give the Europeans a 13-11 win in points.
Russian fifth seed Medvedev has twice thanked a hostile Louis Armstrong Stadium crowd for spurring him to victory, saying their boos had energized him after his second and third-round matches.
Nadal reeled off nine games in succession across the third and fourth sets and 12 of the final 15 to nail down a 40th Grand Slam quarter-final appearance, nine of which have come at the US Open.
World number two Nadal was the only major seed to survive a quarter-final cull after defeats for Dominic Thiem and Alexander Zverev.
Rafael Nadal has now notched the most wins in the elite series, with his 379th Masters victory surpassing Roger Federer by one.
It may be a brief visit to the top five for Tsitsipas with lots of points on offer at this week's Montreal Masters and Kei Nishikori, just five points behind him in sixth.
Less than 24 hours after it was revealed Zverev had split with his coach Ivan Lendl, the world number five wasted two match points before losing to Nikoloz Basilashvili.
Second seed Osaka was joined at the exit door by German sixth seed Zverev and Tsitsipas, the seventh seed, both players tipped as potential champions this year.
The 21-year-old German, ranked fourth in the world, is part of the highly-touted "NextGen" crop of 20-somethings who are making an impact on the ATP Tour.
Zverev joined a list of back-to-back Washington champions that includes Americans Andre Agassi and Michael Chang and Argentina's Juan Martin del Potro.