Top seeds Novak Djokovic, a four-time Rome winner, and defending champion Rafael Nadal, winner nine times, have first round byes in a tournament rescheduled from May due to the coronavirus pandemic. They will begin their campaigns on Wednesday.
Elsewhere 15th seed Dimitrov, a 2014 Rome semi-finalist, saw off Italian wild card Gianluca Mager 7-5, 6-1.
Bulgarian Dimitrov pulled ahead from 5-5 in the first set to win seven straight games and next meets either Yoshihito Nishioka or Kitzbuehel champion Miomir Kecmanovic.
Cilic, who lost to Alexander Zverev in the Rome semi-finals two years ago, battled past Kazakhstan's Alexander Bublik 6-7 (4/7), 6-2, 6-4 for a second round meeting with sixth seed David Goffin.
France's Benoit Paire, who was removed from the US Open after testing positive for Covid-19, lost his opener 6-2, 6-1 to Italian teen Jannik Sinner.
The 19-year-old Sinner, ranked 81, needed just over an hour to advance following a nervy performance by the 24th-ranked French player.
Next Gen ATP Finals champion Sinner will play third seed Stefanos Tsitsipas of Greece in the next round.
US Open quarter-finalist Borna Coric eased past 14th seed Cristian Garin 6-4, 6-4, while Frenchman Ugo Humbert defeated two-time Grand Slam finalist Kevin Anderson 6-3, 7-6 (7/5) in his Rome debut.
He will try to upset seventh seeded Italian Fabio Fognini in the second round.
In the women's tournament, US Open quarter-finalist Elise Mertens showed no problems switching from hard court to clay with the 11th seed disposing of Taiwan's Hsieh Su-Wei 6-3, 6-1.
"My serve was better in the second set, which made all the difference," said the Belgian.
Italian wildcard Jasmine Paolini eased past Latvia's Anastasija Sevastova 6-2, 6-3 to earn a second round clash with top seed Simona Halep of Romania.
Second seed Karolina Pliskova opens her title defence against fellow Czech Barbora Strycova who strolled past Russian Veronika Kudermetova 6-3, 6-3.
Collated results from the first day of the ATP and WTA clay-court tournament in Rome on Monday (x denotes seed):
Filip Krajinovic (SRB) bt Felix Auger-Aliassime (CAN x16) 6-4, 7-5
Casper Ruud (NOR) bt Karen Khachanov (RUS x11) 6-3, 3-6, 6-1
Kei Nishikori (JPN) bt Albert Ramos-Vinolas(ESP) 6-4, 7-6 (7/3)
Stefano Travaglia (ITA) bt Taylor Fritz (USA) 6-4, 7-6 (7/4)
Borna Coric (CRO) bt Cristian Garin (CHI x14) 6-4, 6-4
Lorenzo Sonego (ITA) bt Nikoloz Basilashvili (GEO) 6-3, 6-1
Marin Cilic (CRO) bt Alexander Bublik (KAZ) 6-7 (4/7), 6-2, 6-4
Ugo Humbert (FRA) bt Kevin Anderson (RSA) 6-3, 7-6 (7/5)
Grigor Dimitrov (BUL x15) bt Gianluca Mager (ITA) 7-5, 6-1
Jannik Sinner (ITA) bt Benoit Paire (FRA) 6-2, 6-1
Hubert Hurkacz (POL) bt Dan Evans (GBR) 6-3, 3-6, 7-5
Jasmine Paolini (ITA) bt Anastasija Sevastova (LAT) 6-2, 6-3
Dayana Yastremska (UKR) bt Camila Giorgi (ITA) 7-5, 6-7 (5/7), 6-4
Amanda Anisimova (USA) bt Donna Vekic (CRO x16) 7-6 (7/4), 7-6 (8/6)
Elena Rybakina (KAZ x10) bt Ekaterina Alexandrova (RUS) 6-0, 6-4
Marie Bouzkova (CZE) bt Ajla Tomljanovic (AUS) 6-3, 6-4
Danka Kovinic (MNE) bt Julia Goerges (GER) 6-1, 6-0
Magda Linette (POL) bt Jelena Ostapenko (LAT) 6-4, 6-3
Elise Mertens (BEL x11) bt Hsieh Su-Wei (TPE) 6-3, 6-1
Aliona Bolsova (ESP) bt Alison Riske (USA x13) 5-7, 7-6 (7/4), 6-3
Anna Blinkova (RUS) bt Jil Teichmann (SUI) 6-3, 6-3
Barbora Strycova (CZE) bt Veronika Kudermetova (RUS) 6-3, 6-3