Sharapova turned out for Russia in the Fed Cup earlier this month, but Monday's clash was her first on tour since losing to Serena Williams in Melbourne.
The Russian's Acapulco debut didn't get off to the best of stars, with Sharapova dropping her serve in the opening game of the match.
However, she would recover to break the 22-year-old Rogers to level matters at 3-3 before grabbing another break in the 10th game to take the set.
The second set was a much more straightforward affair for Sharapova winning eight of the last 11 games as she took the set 6-1 and with it the match.
Fifth seed Lucie Hradecka also booked her spot in the second round, but eighth seed Madison Brengle fell at the first hurdle, going down in three sets to María-Teresa Torró-Flor.