Fearnley was ranked outside the top 600 at the start of 2024 but won four ATP Challenger titles to climb inside the top 100 by September.
He has five wins from eight matches on clay – the surface used at Roland Garros for the French Open – since playing his first last month.
Fearnley exchanged breaks of serve with former top-25 player Lajovic in each of the opening four games.
But he took control of the match by breaking for a 5-4 lead and serving out the opening set, before dominating the second.
Fearnley won six consecutive games following Lajovic’s initial hold of serve to wrap up victory in 72 minutes.
Cameron Norrie’s match against Swiss player Dominic Stricker was suspended by rain later on Monday, with the former British number one leading 7-6 (7-2) 3-1.