There's an awesome little find in Tokyo where they specialize in combining the two. You can find delicious gyoza paired with a range of different champagnes at the aptly named Champagne and Gyoza Bar, in the Akasaka area.
Champagne and Gyoza Bar is in a bit of a back-street location, but it's easy enough to walk to from a few different train stations (see the small map below). The best positioned station to get there with the shortest walk is Akasaka, but either Akasaka-Mitsuke or Tameike-Sanno Tokyo Metro lines will have you there in under 10 minutes on foot.
Your gyoza is served up with four different types of dipping sauces, as well as some chili oil and truffle oil. Lately, they've even been doing grated truffles on the gyoza too for that extra luxurious feel! Also, if you're not really feeling the champagne, they also had red wine, white wine and beers on their drinks menu.
If you'd like to head to Champagne and Gyoza Bar during your trip, you can check them out at 3 Chome-15-4 Akasaka, Minato, Tokyo 107-0052. They have an English menu and English speaking staff, too - which definitely makes things easier if you're like me and your Japanese speaking or reading comprehension is minimal. Another bonus is that they accept credit cards, which is perfect if your stash of yen is dwindling from all the fun you've been having!