Ranked 13th in the world after lengthy spells off the court with an elbow injury, Djokovic is training with Vajda for the first time since last year's Monte Carlo Masters.
"I'm very happy as he (Vajda) is more than a coach. He's more like a brother, a father.
"The reality is I've played six or seven matches since July. I'm lacking matches and a bit of confidence on court. But I'm looking ahead."
Djokovic dominated from the outset as Lajovic struggled, and despite some nervy moments on serve in the second set, wrapped up victory in under an hour to book a second-round meeting with Borna Coric.
Djokovic started strongly, dropping only three points on serve as he cruised to a first-set bagel which was sealed with a fierce return winner.
Lajovic had his chances in the second, creating break-point opportunities in three consecutive Djokovic service games as the two-time Monte Carlo champion threw in two double faults.
But the qualifier was unable to make the most of those openings, and Djokovic broke in game four as Lajovic fired wide.
Djokovic had to save two break points when serving for the match, but sealed the win to stay on course for a possible quarter-final meeting with old rival Rafael Nadal.
- Nishikori battles past Berdych -
Earlier on Monday, Japan's Kei Nishikori fought back from a set down to see off Czech 12th seed Tomas Berdych and reach the second round.
In a battle of two former Grand Slam finalists, Nishikori raced through the final two sets to progress with a 4-6, 6-2, 6-1 victory.
But he was close to his best against Berdych, despite making a slow start and calling for a medical timeout for his troublesome wrist.
"It's not great but, I mean, first match on clay, you know, I'm expecting (a) little soreness," said Nishikori.
"Hope I can put more matches on clay. I always enjoy this Europe season. Yeah, hope I can have good preparation these couple tournaments, and hope I can finish great in French (Open)."
The 28-year-old, playing in the tournament for the first time in six years, will take on Russian youngster Daniil Medvedev in round two.
The 21-year-old petulantly insisted a supervisor was called to check the mark, saying: "I can call the supervisor whenever I want." He then sarcastically apologised, saying he was "sorry" to hurt Gabas' feelings.
Ramos-Vinolas, who lost last year's final to Nadal, was leading by a set and a break at the time and went on to win 6-3, 6-3.
Former Wimbledon runner-up Milos Raonic had to come back from losing the first set to Monegasque wildcard Lucas Catarina, while French veteran Julien Benneteau bowed out of his final appearance in Monaco at the first hurdle with a straight-sets defeat by Coric.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)