Jaren Jackson Jr. added 23 points for Memphis, which won its second straight game. Ivan Rabb had a career-high 19 points to go along with 11 rebounds, while Holiday finished with 17 points.
Karl-Anthony Town had 26 points and 18 rebounds for Minnesota, and Dario Saric finished with 22 points. Luol Deng scored 18.
A pair of free throws by Towns with 15.7 seconds left tied it at 106. The game appeared headed to overtime until Holiday was fouled by Josh Okogie in the final second.
Memphis center Marc Gasol was a late scratch amid reports about the Grizzlies being in trade talks with the Charlotte Hornets. The Grizzlies have told teams they are entertaining trade offers for Gasol and Conley in advance of Thursday's trade deadline.
Saric brought a boost off the Minnesota bench to offset an early Memphis lead.
But Conley closed the first half with a trio of 3-pointers in the closing minute to help Memphis to a 66-54 lead at the break.