Raducanu served eight double faults and had a first serve percentage of 51.9%, but the 21-year-old was helped by some impressive ground strokes and a number of errors from Masarova, who is 96 in the world.
“It was super important for me to get through that,” said Raducanu, whose ranking has dropped to 250 an eight-month absence from the tour through injury.
“It’s one of the trickiest tournaments conditions-wise to play in because it varies so much.
“I dealt with that really well and overcame a really tough opponent.”
Raducanu, who reached the fourth round at Indian Wells last year, needed four match points to beat Masarova and set up the second-round tie against Yastremska, who is ranked 31st in the world.
“Tennis, I think, is one of the toughest sports out there about focusing, refocusing,” she added.
“Momentum shifts and the scoring system are pretty brutal so I was really pleased how I stayed really focused and didn’t get down and competed really, really hard in tough situations.”
Elsewhere, American Venus Williams, whose first-round tie against Japan’s Nao Hibino on the opening day was suspended when rain stopped play, was beaten when their match was finished on day two.
Williams won the first set but fell way as she suffered a 2-6 6-3 6-0 defeat in just over two hours, with Hibino reeling off 10 straight games from 3-2 down in the second set.
The 43-year-old Williams hit seven aces but served 10 double faults in her first match since last year’s US Open.