The Indian, seeded 31, beat Tulyaganova 6-4 6-2 in little over an hour long tie to set up a clash with Swiss qualifier Timea Bacsinszky, who beat Australian Sophie Ferguson 6-4 6-3.
"It's not easy to deal with things like that off the court, there are some misunderstandings happening," Sania, who was recently embroiled in a controversy related to insulting the Indian national flag during the Hopman Cup, said after the match.
The 21-year-old, who was also battling minor injuries like an abdominal strain that cut short her season last year, was rusty to start with and wasted a couple of break points besides committing 11 unforced errors in the opener.
But Tulyaganova's patchy form worked to Sania's advantage and the Indian wrested the first set after a 40-minute seesaw battle.
Sania then showed some fine form in the second set. The Indian cut down on her unforced errors broke Tulyaganova twice.
Sania also did well to hold her serve and did not give away a single break point, outplaying Tulyaganova in just 29 minutes to advance to the second round of the season's first Grand Slam.
The Hyderabadi ace said despite the off-court distractions, her focus on the game has not wavered but warned that she was not immune to being affected by the controversies, which have plagued her ever since she started her career.
"... I'm at a Slam and the most important thing is to play tennis. I'm not superhuman so it does affect me a little bit no matter how much I try to block it out, it's still at the back of your mind," she said.
If she gets past the second-round hurdle, Sania, who is placed in the third quarter of the draw, will have to brace up for a possible showdown with eighth seed American Venus Williams.