From Matthew Cederquist, Game Producer for Diablo II:
“Yes! Keep your old save files,” said Cederquist, when asked about a save file importing feature. “Back when we were working on [the remaster], we wondered if the old save files would work, so we kind of shoved it in, and it worked! So yes, your local singleplayer save files will carry over.”
Now I just need to buy a CD drive to access my old backups. And to anyone thinking, "Who still has their backups, sites like this still exist. Never underestimate Diablo II players.