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.
"I think he played a great tournament," Nadal said of his 23-year-old opponent,
"First time on this court (Philippe Chatrier) is never easy, the court is so big. I'm happy with the way I played today, I was able to control the points from the baseline, which is important for me.
"I look forward to playing against David (Ferrer), it is going to be a very difficult match," added Nadal, who ended the masterclass on his first match point with a thumping forehand down the line.
Ferrer is one of three players who have beaten Nadal on his favoured clay this season, prevailing in straight sets in the quarter-finals of the Monte Carlo Masters in April.
It was Ferrer's second victory on the slow surface against Nadal, 10 years after the first one in Stuttgart in their first encounter.
With pop star Prince watching from the stands, Nadal played his part of King of the Clay almost to perfection, a few unusual unforced errors the exception in an otherwise extremely solid game.
Lajovic, the man who lost an Australian Open qualifier two years ago after blowing a 6-2 5-0 40-0 lead, surrendered all hope after just a few games.
Nadal, who turns 28 on Tuesday, powered to a 5-0 lead, allowing his opponent only five points before slowing down a bit, only for the Spaniard to step up another gear at the beginning of the second set.
A series of winners - most of them forehands - combined with Lajovic's catalogue of unforced errors, left the Serbian staring at a possible bagel.
His fine backhand helped him avoid the humiliation and even gave him a break late in the third set, but Nadal, who did not seem troubled by his back pains, had already turned his mind towards Wednesday's quarter-final against Ferrer.
(Editing by Ed Osmond)
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