The unseeded Belarusian Azarenka broke Krejcikova four times, saved three of five break points and won the final five games of a match lasting one hour 45 minutes.
Krejcikova battled to get the opener back on serve but Azarenka broke right back and then went on to serve out the first set in the following game on her second set point.
In the second set, Krejcikova grabbed the early momentum and it was Azarenka who had to come back from a break down and then kicked into a higher gear before saving two break points in the final game to serve out the match.
"I am just a fighter and I will keep going until I get it right," Azarenka said during her on-court interview.
"My mentality has been like this since I'm a kid. Growing up not in the greatest conditions I had to fight for everything and that really shaped my personality today."
Wozniacki, who reached the second round last week in Montreal where she played her first competitive matches after more than three years away to start a family, fell 6-4, 6-4 to lucky loser Varvara Gracheva of France.
Former world number one Wozniacki had her chances but failed to convert any of her nine break points while Gracheva's backhand proved too much for the Dane to handle.
Czech ninth seed Petra Kvitova, a Cincinnati finalist last year, also booked her spot in the second round with a 7-6(2), 6-0 win over Russian Anna Blinkova.
In other early action, Jelena Ostapenko beat former Cincinnati champion Karolina Pliskova 1-6, 6-3, 6-4 and will next face fourth seed Elena Rybakina.
Tunisian fifth seed Ons Jabeur, in her first match since suffering "the most painful loss" of her career in the Wimbledon final, is scheduled to cap the day session on Centre Court when she meets Ukraine's Anhelina Kalinina.
Holder Carolina Garcia of France will kick off the evening session against 2017 US Open champion Sloane Stephens.
