Alcaraz energised by playing for the flag after seeing off Paul
The world number one prevailed 6-3, 7-6(7) to reach the semi-finals on his Games debut, but in cloying humidity he had trailed in the second set, looking flat and frustrated.
It was hardly surprising, his seventh match in six days coming came the day after an emotional doubles defeat with his idol Rafael Nadal in what could have been the latter's last Roland Garros match.
Second seed Alcaraz was kicking the hoardings in frustration at one point as he trailed 5-3 with Paul bossing the match.
He snapped out of his slump, however, and is now two wins away from adding the Olympic gold to a spectacular year which has brought him the French Open and Wimbledon titles.
"I was trying to raise the adrenaline and the intensity and the truth is that it helps to know that you playing for Spain, we are playing and representing our flag," Alcaraz told reporters.
"That helps a little more to leave everything on the court, to continue fighting, to keep running and not give up."
Alcaraz hit the ice bath after beating Paul and will need a quick recovery for his semi-final against either Canada's Felix Auger Aliassime or Norway's Casper Ruud.
"Singles, doubles, not a day of rest in between, but the Olympics are every four years and you have to accept that," the four-times Grand Slam champion said.
"You have to accept it, you have to fight an try to do everything perfect off the court."
Even when not playing at his highest level, Alcaraz's presence on court can make the difference.
Ninth seed Paul will look back at pivotal moments, not least the set point he had in the second set tiebreak when he did not do enough with a half volley and Alcaraz sprinted across court to punch a backhand down the line.
"It was a honestly a difficult moment in the second set but representing Spain helps a lot to increase the level," he said.
"Tommy played great tennis but I'm happy I found solutions."