Nadal was tested by the spirited Serbian in the opening set, but went on to wrap up the match in less than two-and-a-half hours with a 7-6 6-2 6-2 victory.
It looked as though the two-time champion might be in trouble early on at Flushing Meadows when Lajovic registered an early break before earning the opportunity to serve for the opening set at 5-4.
However, the world number one broke back with a love game to force a tiebreak, which he eventually won 8-6 to put himself in control of the match.
It was more straightforward from that point on, with the Spaniard breaking three times to build a 4-1 lead in the second set before going on to claim it 6-2.
The number one seed broke again in the opening game of the third to move closer to the second round, and another break later in the set helped him close out the victory with his first match point.
Nadal will face either Taro Daniel or Tommy Paul in the second round.