Young, who finished with 19 points, connected on a trio of 3-pointers within a 1-minute span as the Clippers erased a 27-point second-half deficit and scored an improbable victory in the first game of their Western Conference quarterfinal series.
After Rudy Gay's 10-footer gave the lead back to Memphis, Paul (14 points, 11 assists) sank a pair of go-ahead free throws with 23 seconds remaining.
Gay's desperation jumper missed at the final buzzer, sealing the Clippers' amazing comeback.
Blake Griffin supplied 17 points and seven rebounds in the victory.
Gay had 19 points and seven boards for Memphis, which was held without a basket for nearly 8 minutes in the fourth quarter after building an 84-57 lead.
The Grizzlies will try to bounce back in Wednesday's Game 2 in Memphis.