Alcaraz, Sabalenka cruise to wins at the Italian Open
World number three Alcaraz got past Lajovic in one hour and 23 minutes in his first match since suffering the injury in his Barcelona Open final defeat to Holger Rune last month.
The Spaniard showed some signs of ring rustiness with 22 unforced errors but still managed to win the first four games of the match to give Lajovic too much to do in the first set.
And Alcaraz closed out the match in professional style with another quick start in the second set which put him three games ahead, allowing him to serve out for the win.
"Today was a really good day in the office, hopefully tomorrow and Sunday is going to be better," said Alcaraz.
"Of course I tried to make the most of my time, when I'm not playing tournaments I always try to find the positive things... I could stay at home more with my friends, with my family, recovering and getting a little bit fresh mentally.
"It was a great performance, a great level, which was surprising for me a little bit but I'm just really happy for that."
Alcaraz will now face one of Alex Michelsen and Laslo Djere in the third round as he eyes a potential final with world number one Jannik Sinner in the Italian capital.
Sinner will make his comeback from a three-month doping ban on Saturday when he takes on Mariano Navone in front of what will be a partisan crowd at the Foro Italico.
Alcaraz later told reporters that he would take it easy on his day off, keep an eye on Sinner's match and go to watch his friend Patric play for Serie A football club Lazio against Juventus at the nearby Stadio Olimpico.
Sabalenka through
One of Alcaraz's potential opponents in the last 16, Grigor Dimitrov, fell at the first hurdle against wild card entrant Francesco Passaro.
Bulgarian Dimitrov, the 14th seed, was beaten 7-5, 6-3 by Italian Passaro, who will take on Roman Burruchaga or Karen Khachanov in the next round.
Last year's winner Alexander Zverev is in action, against Argentina's Camilo Ugo Carabelli, in the last match of the day on centre court.
Home hope Lorenzo Musetti opened his week, his first in the top 10 of the men's world rankings, with a 6-3, 6-2 win over qualifier Otto Virtanen.
In the women's draw, Aryna Sabalenka cruised past Anastasia Potapova and into the third round with a 6-2, 6-2 victory.
Last year's beaten finalist and current world number one, Sabalenka swept past Russian Potapova in just over an hour.
"I'd say that the top player is always a favourite," said a confident Sabalenka.
"I know that if I bring my best game and my fight spirit on court, I know that I can win this tournament."
Sabalenka will play Sofia Kenin in the next round after the American beat another Russian in Anastasia Pavlyuchenkova in straight sets, 6-3, 6-0.
The 27-year-old Sabalenka has reached the final of her last four tournaments, with 1000 series wins in both Miami and Madrid.