Former world number one Azarenka, who travelled to the Stuttgart event from Australia, won 7-5 6-4 in round one.
The Belarusian said the effects of her flight proved a “huge challenge”, adding: “I’m very happy with the way I handled it.”
Azarenka, 29, now faces defending champion Karolina Pliskova.
Russian fourth seed Pliskova – the world number four – is among four of the world’s top five players featuring in the event.
Czech world number three Petra Kvitova, who received a bye to the second round, cruised through to the quarter-finals with a 6-1 6-4 win over Belgian qualifier Greet Minnen.