Played on the hottest opening day in Wimbledon history, the match was stopped at 3-0 in the final set when a fan fell ill.
Alcaraz rushed over with a water bottle and the contest was stopped for more than 15 minutes before resuming.
“I think battling the heat is one of the things that we have to do,” Alcaraz said in his post-match press conference. “Probably the match can’t be in another way, because of the heat. I think we have to deal with it in the best way possible.
“Obviously for the fans, and even here in London, they are not used to having this weather. Sitting five hours in the stand without moving, the sun hitting, I think could be really difficult to deal with for the spectators.
“But we are playing (an) outdoor tournament, so I think that the roof shouldn’t be closed because of it.”
Alcaraz looked out of sorts for long periods against 38-year-old Fognini, although the reigning champion took the first set 7-5 after saving several break points.
He might have expected to kick on after taking the lead, but Fognini was in an obdurate mood and matched the world No. 2 in an equally close second set.
This time, though, the Italian edged it via a tie-break to level things up.
Another set of twists and turns went Alcaraz’s way 7-5 and in the heat there were suggestions that Fognini might wilt.
Instead, he produced a superb performance in the fourth set, twice breaking Alcaraz to take it 6-2 and leave the Spaniard fearing an unexpected and premature exit.
But the two-time winner roused himself during the break, stepping things up in the final set and blowing his opponent away.
He won the first five games to alleviate any worries of a shock result, and then served out the set to win and set up a second-round clash with British qualifier Oliver Tarvet.
“I just told him that I have big respect to him,” Alcaraz said of his exchange with Fognini at the net.
“He’s the best. Everything that he has done in his career, what he has done today. I love watching him play.
“I’m just glad to have shared the court once again with him, that I have shared the locker rooms, the tournament with him. Just amazing. Just have big respect to him.”
Alcaraz is bidding to become the fifth man in the Open Era – after Bjorn Borg, Pete Sampras, Roger Federer and Novak Djokovic – to win the Wimbledon men’s singles title three years in a row.
