Fearnley was classified outside the top 600 in the beginning of 2024 but won four ATP competition titles to climb it in the best 100 in September.
He has five wins from eight games on Clay – the surface used in Roland Garros for the French Open – since he played last month.
Fearnley exchanged break with the former Lajovic players in each of the four opening matches.
But he dominated the match by breaking the progress of 5-4 and offering the opening group, before he dominated the second.
Fearnley won six consecutive matches after the first Lajovic contract of finishing victory in 72 minutes.
Later on Monday, Cameron Nuri is facing the Swiss player Dominique Strch after the previous British number came through the qualifiers.
