A six-time winner in Rome, Djokovic was made to work by the Argentine, prevailing in one hour, 51 minutes to set up a third-round clash with Grigor Dimitrov who beat Stan Wawrinka earlier in the day.
Djokovic has not been knocked out of an ATP tour event at the first attempt since the Monte Carlo Masters last year, just his second tournament last season following his refusal to get vaccinated against Covid-19.
Still nursing an injury to his right elbow he had to battle his way past Etcheverry, winning the first set via a tie-break before finally ensuring passage into the next round.
"I've said before that this surface requires more time for me than maybe for other players to get myself to a good level, move well and hit the ball well. Rome has always been a tournament that I need for Roland Garros," Djokovic said in Italian to Sky Sport.
"This is my favourite clay tournament, there's an incredible atmosphere on centre court and the consistency that I've had in previous years gives me a lot of faith."
Djokovic has made 12 finals in Roma and said on Thursday that he was feeling good ahead of the tournament despite missing last week's Madrid Masters.
Wearing an elbow brace he made a slow start, dropping his serve in the first game and struggling to deal with the Argentine's powerful shots.
The first set hinged on a scrappy game six, in which Djokovic broke to level the scores following a battle at deuce.
From there, the set went with the serve until Djokovic won the final four points in the tie-break to take the lead.
Djokovic then kicked up a level, breaking serve in the first and seventh games of the second set to give the match a more straightforward look than Etcheverry's performance deserved.
Perfect Swiatek
Iga Swiatek started her bid to win a third straight Italian Open title in perfect fashion, not dropping a game in her straight-sets thrashing of Anastasia Pavlyuchenkova.
World number one Swiatek looked in exceptional form as she demolished Pavlyuchenkova 6-0, 6-0 in just over an hour to ease her way into the third round in Rome.
"I still may feel some discomfort, but it is really, really low. Even when I withdrew from Miami, the pain was low. It was just the risk of getting it worse that stopped me," Swiatek told reporters.
"I wouldn't say my physicality is stopping me. I came back quickly and I kind of missed only one tournament. I think overall it's pretty positive."
Swiatek now faces either Lesia Tsurenko in the last 32 after extending her winning streak in the Italian capital to 12 matches.
The only real resistance from Pavlyuchenkova, who reached the final of Roland Garros two years ago, came in the final game of the first set, when she saved six set points from 0-40 down on her serve.
Swiatek eventually came through that game and swept Pavlyuchenkova aside in the second set, finishing the job against the serve with her first match point.
Fourth seed Ons Jabeur fell 6-1, 6-4 to Paula Badosa, last year's finalist being eliminated in her first match to leave the women's tournament without another big name.
Jabeur joins world number two Aryna Sabalenka, third-ranked Jessica Pegula and 2019 winner Karolina Pliskova in being dumped out in the second round.
The 28-year-old has had calf problems in recent weeks, an injury which caused her to retire in the semi-finals at Stuttgart and not defend her Madrid title.
Badosa will face world number 27 Marta Kostyuk in the third round.
Sinner the winner
Earlier, world number eight Sinner took one hour and 18 minutes to deal with Australian qualifier Kokkinakis, winning 6-1, 6-4, and will play Alexander Shevchenko, who beat Sebastian Baez 6-3, 6-4, in the next round.
"I'm happy with my level today, it was not easy, it was a little bit breezy, a bit windy," Sinner said on court.
"Especially in the second set he played a little bit better. I also had a couple of chances which I didn't use but very happy about how I served, I struck the ball very well."
Sinner did not drop a single point on his serve in a dominant first set in which the gulf in class between the Italian and Kokkinakis was painfully evident.
Kokkinakis, ranked 104 in the world, put up more of a fight in the second set but once Sinner broke in the seventh game to take the score to 4-3, it was only a matter of time before the deal was sealed.
Sinner made no mistake, serving to love in games eight and nine to ensure comfortable passage to the next round.
Rome has not seen an Italian winner of the tournament since Adriano Panatta in 1976.
With local boy Matteo Berrettini out of action again with abdominal problems, Sinner will be getting the bulk of the capital city's support.
"It is a very special feeling especially here in Rome," Sinner added.
"You know trying to be an example for the young kids, also spending a little bit of time with them. It's important to show them the love."
Sinner has had a decent season far, winning in Montpellier in February and losing the finals in Miami and Rotterdam to Daniil Medvedev.
He also reached the semi-finals at Indian Wells, losing to eventual winner Carlos Alcaraz.
Cameron Norrie strolled past French qualifier Alexandre Muller 6-2, 6-3 to set up a clash with Marton Fucsovics, who upset Alex de Minaur 6-3, 6-4, in the next round.
Muller had won his first-ever main draw match in a Masters tournament in the first round, beating another Briton in Kyle Edmund, and was no match for World number 13 Norrie.
