Paul's foul shot with 18.3 seconds left gave the Clippers a two-point advantage at 89-87, and Crawford iced the contest with two more free throws with 2.2 seconds to go.
Paul finished with 13 points and five assists.
DeAndre Jordan contributed 16 points and 12 rebounds as the Clippers recorded their fifth straight win, strengthening their grip on the Western Conference's fourth playoff seed.
Marc Gasol and Zach Randolph had double-doubles for Memphis, whose three-game winning streak ended. Gasol had 18 points and 15 rebounds for the fifth-seeded Grizzlies and Randolph added 16 points and 12 boards.