Shapovalov was beaten by fellow Canadian Vasek Pospisil in his opening match at the Oden Sud de France last week after falling at the first hurdle at the Australian Open.
The world number 16 suffered another setback, going down 6-3 7-6 (7-3) to the unseeded Dimitrov.
Dimitrov was beaten by outsider Gregoire Barrere in his first match in Montpellier last week but claimed a fourth tour-level victory of the year after saving all three break points he faced against the eighth seed.
The former world number three could face another Canadian in the form of Felix Auger-Aliassime or German Jan-Lennard Struff in the second round.
Roberto Bautista Agut, the sixth seed, came from a set down to defeat Marton Fucsovics, while fellow Spaniard Pablo Carreno Busta downed Adrian Mannarino 7-5 6-7 (3-7) 6-4.
Dan Evans advanced in the final match of the opening day of the tournament, seeing off qualifier Philipp Kohlschreiber 6-3 7-5.