This is just a second match on grass for Alcaraz after the physical and emotional toll of the Roland Garros final.
He started strongly against Munar but grew unusually frustrated as the match went on, shouting at his team and disputing a time violation with the umpire midway through the second set.
World number 59 Munar said before the match that his chances of an upset were "low" - but he came within touching distance of victory, thanks to some strong serving and deep, powerful forehands.
After a composed first set, Alcaraz could not find the breakthrough in the second, with Munar saving three break points in a gritty display.
Munar also missed nine break points and needed eight set points to eventually edge over the line after one hour and 41 minutes.
Alcaraz held to love to start the deciding set and quickly went a break up, with Munar looking to have expended all his energy taking the second set.
However, Munar broke back and led 4-3 30-0 before Alcaraz hit back, reeling off four points in a row to force the set back on level terms.
There was a glimpse for Munar as Alcaraz went 0-30 down in his next service game - but, as the top players often do, Alcaraz simply found a way through, holding serve before capitalising as Munar served for a match tie-break.
"It was a really tough battle. He's a great competitor and showed how difficult it is to beat him," Alcaraz added.