The Belarusian was far from impressive in the opening game, managing to hold after six deuces before finding her feet on Number One Court.
Azarenka broke three times in the opener to take the set 6-1 in just 36 minutes.
Another break at the start of the second gave the world number two an early two-game advantage in the set.
She broke the American's serve again in the fifth game to take a 4-1 lead but was this time pegged straight back by Falconi.
The pair then exchanged breaks of serve late on, but it was Azarenka who held to secure the match.
Azarenka will next face Italy's Romina Oprandi on Thursday at the All England club.