Six of the world's top 10 are expected to play in the hard-court tournament, the final WTA 1000-level event this season before the WTA Finals, with Polish world number one and US Open winner Iga Swiatek (21) absent after winning in San Diego on Sunday.
Azarenka, who reached the last 16 in New York but lost in her Ostrava opener earlier this month, dropped only three first-serve points in the first set and broke in the seventh game.
The Belarussian's serve deteriorated in a choppy second set but her opponent could not derail the experienced Azarenka who set up a second-round meeting with Spanish top seed Paula Badosa (24) in a repeat of the Indian Wells final last year.
Canadian Andreescu had no problem overcoming Teichmann, winning the first three games of the first set and cruising through the second as the Swiss served three double faults.
Russian Liudmila Samsonova (23), who won in Cleveland and Tokyo this year but lost her first-round match in San Diego, beat Estonian Kaia Kanepi (37) 6-1 7-5 and China's Zhu Lin (27) defeated Frenchwoman Alize Cornet (32) 3-6 6-3 6-4.
The biggest upset of the day belonged to Colombian Camila Osorio (20), who broke 15th seed, Ekaterina Alexandrova (27), nine times en route to a 6-4 7-6(8) victory.
American Sloane Stephens (29), Belgian Elise Mertens (26), Ukrainian Marta Kostyuk (20) and Canadian Rebecca Marino (31) all advanced to the second round after straight sets wins.
