Two conversions from four break points were enough for 21-year-old Spaniard to take the first set, one from one handed him the second, before the experienced Portuguese number one snatched the momentum back at the end of the third.
Borges blew one set point at 6-5 by going along with a return but made no mistake in the tiebreak to claim the set and leave Alcaraz seething in his chair.
The four-times Grand Slam champion came out in the fourth set like a man on a mission, breaking at the first opportunity for 2-0 with a spectacular over-the-shoulder pass from the back of the court.
A huge roar of “Vamos!” echoed over the sunbathed Rod Laver Arena and from that point Alcaraz’s place in the fourth round for the 11th time in 16 Grand Slam campaigns looked assured.
The remainder of the set from Alcaraz was venomous serving, a few sumptuous drop shots and the customary barrage of forehand winners, the 35th of which brought up match point after just under three hours on court.
“I’m just really happy to be able to play here once again and show my best tennis here. It’s been a pleasure whenever I step on this court, it’s a beautiful court. The last time I played here, I lost so I really wanted a win here,” Alcaraz said.
He already has two Wimbledon crowns and one title each at the U.S. and French Open so the Australian Open, where his best result was his quarter-final appearance last year, would count as his least successful Grand Slam.