Eight-times champion and world No.1 Nadal improved his Roland Garros win-loss record to a remarkable 63-1 to advance having not dropped a set in Paris this year.
World No. 83 Lajovic, who had not previously lost a set on his French Open debut, quickly went 5-0 down and never recovered, also losing the first four games of the second set to love.
Nadal ended the masterclass on his first match point with a thumping forehand down the line.
Results from the French Open men's singles matches on Monday
1-Rafa Nadal (Spain) beat Dusan Lajovic (Serbia) 6-1 6-2 6-1
5-David Ferrer (Spain) beat 19-Kevin Anderson (South Africa) 6-3 6-3 6-7(5) 6-1
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