American Kenin, who was once ranked number four but has slipped to 134 in the world following her 2020 Melbourne Park triumph due to illness and injuries, was far from her best but managed to get past Bucsa in a little over three hours.
"It was obviously tough, but I was able to somehow get through. I didn't feel a 100% but it doesn't matter, a win is a win and I'm going to take it," said Kenin, who converted seven of her 21 breakpoints in the match.
"It could have gone either way and it was a matter of who could be mentally strong. I felt tired and... after three hours I'd imagine that she was too, so I just had to battle it out."
Former US Open runner-up Leylah Fernandez (20) was beaten 7-6(4) 4-6 6-3 by Aliaksandra Sasnovich (29), while 2019 French Open runner-up Marketa Vondrousova (23) overcame Kaia Kanepi (37) 6-0 4-6 6-2 in other first round matches.
Earlier, former doubles world number one Barbora Strycova (37) downed Maryna Zanevska (29) 6-1 3-6 6-3 for her first singles victory since 2020 following her return to the WTA tour at last week's Madrid Open from a two-year maternity break.
The Czech will take on world number eight Maria Sakkari (27) in the second round.
"I can't believe it," Strycova, who celebrated her victory with her son Vincent, said. "I don't know how it feels because I didn't really process it.
"I went off the court and right away grabbed my son because he was hungry. So I wasn't really realising I won the match.
"I'm very happy. I'm a little shy about being happy. Also, I feel a little strange because he's here and I will be changing a diaper in a little while instead of getting a glass of wine."
Mayar Sherif (27), who became the first Egyptian ever to reach a WTA 1000 quarter-final in Madrid, crashed at the opening hurdle against Russian Anna Blinkova (24) who prevailed 2-6 6-2 6-3.
The men's tournament headlined by six-times champion Novak Djokovic (35) and Rome debutant Carlos Alcaraz (20) begins on Wednesday.
