Norwegian Casper Ruud (23) survived a tight first set to push past Slovakian Alex Molcan (24) 7-6(3) 6-3.
Alcaraz had the momentum after he bagged the first-set tiebreak before opening up a 4-1 lead in the second.
But the 34th-ranked American clawed his way back, converting break points in the seventh and ninth games.
Alcaraz stood on the cusp of victory when he earned a match point during the second set tiebreak but once Paul saved it, the 19-year-old Spaniard's challenge faded as he struggled to overcome the disappointment of not wrapping up the match in straight sets.
The American, who won his sole ATP title at the Stockholm Open last year, fired off 14 winners in the third set.
Although Alcaraz saved four match points in the decider, he could not deny Paul from completing one of the biggest wins in his career.
Paul clinched it at the net in the next game, offering up a subdued celebration.
Ruud had little trouble with Molcan, firing off 13 aces and 40 winners to set up a clash with Spain's Roberto Bautista Agut.
