The 19-year-old Russian fifth seed, who triumphed at Roland Garros this month, used her big serve to great effect with eight aces but she also produced seven double faults as she tried to outgun the experienced Linette, 15 years her senior.
Linette produced some delicate dropshots and sharp volleys to test Andreeva as the players traded service breaks in both sets, but the Russian hit two unreturnable serves to take the first set and wore down her opponent in the second, serving out for the win.
"It's just always super special to get past the first round at a Grand Slam because you always have that extra pressure," she told reporters.
"For me, I was also thinking that it would be very important for myself to just kind of prove myself that even if I did well at a last Grand Slam, that I can still continue to try and do well."
Linette was not afraid to come to the net and produced delicate dropshots and sharp volleys to test Andreeva as the players traded service breaks in both sets, but the Russian hit two unreturnable serves to take the first set and wore down her opponent in the second, serving out for the win.
Andreeva is bidding to become the first player to win back-to-back French Open and Wimbledon titles since Serena Williams in 2015.
She faces a tough second round battle with 2024 Wimbledon champion Barbora Krejcikova, and said she would be discussing her game plan with coach, another former Wimbledon champion, Conchita Martinez.
"She (Krejcikova) is a two-time Grand Slam champion. She's very experienced. She's playing well on grass. So for sure it's not going to be an easy match for me," Andreeva said.