Bouchard, the No. 6 seed from Westmount, Que., had six service breaks against Diyas, who is ranked 34th in the world.
Both players had trouble with their serves. Bouchard put only 65 per cent of her first serves in play, and won just 62 per cent of those points.
Diyas put 71 per cent of her first balls in play, but won only 21 of 45 points from those serves.
Bouchard will face either No. 10 Carla Suarez Navarro or Heather Watson in the third round.
After a breakout season in 2014 that saw her reach the semifinals of three Grand Slam tournaments, Bouchard has struggled this year.
She hadn't won since reaching the fourth round at Indian Wells in March. She followed that by losing her first match in her next three WTA Tour events, as well as both her singles matches in Canada's Fed Cup loss to Romania in April.
Elsewhere, Christina McHale of the U.S. defeated last year's finalist, Sara Errani, 6-4, 6-4 in front of the Italian's home crowd and will next face top-ranked Serena Williams.
Also, former No. 1 Victoria Azarenka eliminated fifth-seeded Caroline Wozniacki 6-2, 7-6 (2), second-seeded Simona Halep beat Alison Riske of the United States 6-3, 6-0, Russian qualifier Daria Gavrilova eliminated seventh-seeded Ana Ivanovic 5-7, 7-6 (2), 7-6 (7); and Serbian qualifier Bojana Jovanovski defeated Madison Keys of the U.S. 6-4, 1-6, 6-3.
Venus Williams, the No. 14 seed and 1999 champion, beat Elina Svitolina 6-4, 6-3.
In men's action, Andy Murray improved his record on clay this year to 10-0 with a 6-4, 6-3 win over 38th-ranked Jeremy Chardy.
The third-seeded Murray considered withdrawing before the tournament because of fatigue after his perfect start to the clay-court season, but his game showed no signs of wear at the Foro Italico.
"My legs were a bit tired at the beginning then I started to feel better after the first few games. I felt like I played a really great match," said Murray, who committed only 10 unforced errors to Chardy's 26. "I moved well and made it difficult for him to get free points."
Seven-time Rome champion Rafael Nadal was also sharp in a 6-2, 6-0 victory over Turkish qualifier Marsel Ilhan; and Roger Federer began his pursuit of an elusive crown with a 7-6 (3), 6-4 win over 24th-ranked Pablo Cuevas of Uruguay.
Rome is one of only two Masters titles that Federer has not won — along with Monte Carlo. He's a three-time finalist.
Federer often followed his serve to the net and displayed a series of crisp volleys.
Also, fifth-seeded Kei Nishikori advanced with a 7-6 (3), 7-5 win over Jiri Vesely of the Czech Republic.
Fabio Fognini, Italy's top-ranked man at No. 31, upset 10th-seeded Grigor Dimitrov of Bulgaria 7-6 (9), 4-6, 6-0 and will next meet sixth-seeded Tomas Berdych.
Murray has never been past the semifinals in Rome and is looking to peak at the French Open, which starts in less than two weeks.
"It's not impossible," Murray said of his chances of reaching the final or winning at Roland Garros. "Whereas, maybe the last few years, even though I played some good tennis in Paris I didn't feel like I was maybe healthy enough, or played enough matches or had enough big wins against top players to believe I could do it.
"Maybe this year will be different. There's a chance. If I play well then I know I have the chance to go far in the tournament."
Nadal was never really tested against the 87th-ranked Ilhan, who he faced for the first time.
Nadal dropped out of the top five in the rankings this week for the first time in 10 years, and acknowledged he was yet to put together two solid weeks on clay this season.
"But today was a good start," the Spaniard said. "I played with not many mistakes and that gave me the chance to win with that score."
Nadal will next meet John Isner, who served 11 aces in a 7-6 (6), 6-4 win over Leonardo Mayer of Argentina.
Nadal is 5-0 against Isner but needed three sets to beat the six-foot-10 American in Monte Carlo last month, and had to come back from two sets down against him during the first round of the 2011 French Open.
"He's difficult to play against," Nadal said. "You dont have a lot of control on the things that are going to happen. If he serves well then you are in trouble. I have to play with no mistakes when I'm serving."
— With files from The Associated Press