Playing for the first time since retiring from the WTA event in Wuhan, China in September, Azarenka broke Vesnina six times to wrap up victory.
She will have a much sterner test against former French Open finalist Halep, who had a first-round bye.
Australian Sam Stosur had a much difficult match and narrowly avoided disappointment on home turf by battling through a wrist injury to claim an opening victory.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
The former U.S. Open champion recovered from 0-3 down in the final set to beat Slovakian qualifier Jana Cepelova 6-4 3-6 6-4 to start her year.
The 31-year-old has frequently failed to hit the heights on home turf having never won a tournament or progressed past the fourth round at the Australian Open.
The world number 27 looked set for another disappointment before a spirited effort saw off the world number 147 and set up a second-round clash with Spain's Carla Suarez Navarro.
"I am really pleased to walk away with a win, especially being down in that third set and then find that ability to come back and play well and win it when you are down, to turn things around," Stosur said.
Italy's Roberto Vinci, who ended Serena Williams's bid for a calendar grand slam in the U.S. Open semi-finals last year, beat former world number one Jelena Jankovic 3-6 6-2 6-4.
Russian Maria Sharapova is the tournament draw card and she will kick off her campaign on Tuesday against compatriot Ekaterina Makarova. On the men's side, 17-times grand slam champion Roger Federer will play Germany's Tobias Kamke.
Bulgarian Grigor Dimitrov, who has slumped from eighth to 28th in the rankings, started the year with a 6-3 7-6(10) win over fifth seed Gilles Simon of France.
(Writing by Patrick Johnston and Sudipto Ganguly; editing by Ed Osmond)
