The 2014 US Open champion came out of the blocks quickly, breaking Fabbiano's serve in the third game of the match and though he was pegged back to 3-3, he immediately moved back into the lead and reeled off four straight games to seal the first set and move ahead in the second.
Highlights - Cilic comfortably beats Fabbiano
Fabbiano had the chance to serve for the second at 5-4 but Cilic rallied to sneak it 7-5 with his third set point. Fabbiano again looked to get on the front foot with a break in the first game of the third, but Cilic then turned on the style, rattling off six straight games and dropping just two points in the final four.
Elsewhere in the men's draw, 2014 semi-finalist Ernests Gulbis is out after succumbing to Casper Ruud 6-2, 7-6(2), 6-0, continuing his poor run of form, with the Latvian having won just one main-draw match since February.
“I would have wanted obviously to play few more matches in these tournaments. I had a couple of OK tournaments and a few losses,” Cilic said. “But in this week, I focused a lot just to have good intensity in the training, to have good focus. It's important when you come to the Grand Slams that you are focused every single session, that you can keep going on like that.
Elsewhere in the men's draw, Grigor Dimitrov had to go the long route to beat Janko Tipsarevic. Having stormed into a two-set lead, the Bulgarian was pegged back to 2-2 before claiming the decider 6-4.
Kei Nisikori had no such trouble, overcoming Frenchman Quentin Halys in straight sets 6-2, 6-3, 6-4. Marco Cecchinato, a semi-finalist last year, crashed out in the first round, squandering a two-set lead against home hope Nicolas Mahut 2-6, 6-7(6), 6-4, 6-2, 6-4 on the newly-minted Court Simonne Mathieu.
27th seed David Goffin beat Lithuanian Ricardas Berankis in straight sets, 6-0 6-2 6-2 in an easy victory.
Mahut roars back to win five-set duel
FRENCH OPEN MEN'S SINGLES ROUND 1 RESULTS
M. Cilic (Cro)  def. T. Fabbiano (Ita) 6-3, 7-5, 6-1
C. Ruud (Nor) def. E. Gulbis (Lat) 6-2, 7-6(2), 6-0
A. Popyrin (Aus) def. U. Humbert (Fra) 3-6, 6-3, 7-6(10), 6-3
S. Tsitsipas (Gre) def. M. Marterer (Ger) 6-2, 6-2, 7-6(4)
O. Otte (Ger) def. M. Jaziri (Tun) 6-3, 6-1, 4-6, 6-0
G. Dimitrov (Bul) def. J. Tipsarevic (Srb) 6-3, 6-0, 3-6, 6-7(4), 6-4
M. Berrettini (Ita) def. P. Andujar (Esp) 6-7(3), 6-4, 6-4, 6-2