It will be the second final of the season for Djokovic, who began the year by winning the Australian Open.
Nadal seemed to have the momentum after winning 13 out of 16 points at the end of the second set but he spurned two break points in the decider before Tsitsipas broke twice for 5-2.
"I played the best match of the clay-court season for me", said Djokovic, "against arguably the best tennis player in the world on this surface".
"He knew I had a tough match last night, so he took advantage...[made] me run and suffer more and I just didn't have solutions".
The 18-year-old Auger-Aliassime is having his breakthrough season on tour, having moved to 30th in the rankings after reaching the final in Rio and the semifinals in Miami.
Overall Nadal looked reasonably sharp all match, serving well and moving quickly across the court against a player who is just 18 years of age.
Nadal insists that what's happening to him of late on clay is more normal than was his dominance on those courts for so many years.
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Nishikori and Wawrinka will meet in the third round on Thursday, an early showdown for two players that have reached 12 Grand Slam semi-finals between them.
Djokovic further said that Nadal's records in Paris are commendable, asides stating that he is always the ultimate challenger while playing on clay. "I played some good tennis and will be No 4 in the world on Monday, it is just all fantastic things", Bertens said.
"I struggled a lot to get fit and recover that energy, which I now feel again".
"I feel happy with my performance, but at the same time, I feel empty", said Tsitsipas.
21-year-old Tiafoe is now the second highest ranked American player behind John Isner, but the fact that is only 37 in the world shows how far American men's tennis has fallen behind.
"After Indian Wells, I had a big mental drop", Nadal said. This is the first time since 2004 that Nadal arrived in Madrid without a title.
"I needed to step up, Stefanos is very talented". Nine of Djokovic's last 11 victories have been decided in straight sets. After saving match points in the previous round, against Gael Monfils, Federer spurned several chances in the second set tiebreak to progress to today's semi-finals before Thiem edged the decider.
The seventh-ranked Dutch became the first woman to win the Madrid title without dropping a set.