Moreno scored on a half volley from 22 yards off a wild scramble for the ball in the 95th minute.
While the Timbers (11-8-13, 46 points) had a four-game winning streak end, the valuable point moved them to fifth in the Western Conference, three points above the playoff line with two games left.
Columbus (9-7-15, 42 points) took a 1-0 lead on Kevin Molino's goal in the 36th minute but have dropped 19 points from winning positions, including 14 in the past 13 games.
They have also given up a league-high 15 goals after the 75th minute, including three in the previous two games that cost them a potential win at CF Montreal and a tie at Inter Miami SC.
Columbus holds the last playoff position in the Eastern Conference by two points over Atlanta United.
The Crew used a crafty combination to take the 1-0 lead. Pedro Santos found Lucas Zelarayan at the top of the box. Zelarayan drew the attention of several defenders so he dished a pinpoint to catch Molino on stride on the left for his shot.
Molino has battled knee injuries in his two seasons in Columbus. The goal was his first of the season but 34th in his 101st MLS start. He last scored Aug. 4, 2021.
Crew goalkeeper Eloy Room had a quiet first half but made back-to-back saves at the hour mark. The first was from a one-timer by Yimmi Chara directly in front of him, the second was a header by Bill Tuiloma off the ensuing corner kick.
The Crew momentarily thought it had a two-goal lead in the 81st minute but Derrick Etienne Jr. was ruled offside before he put the ball in the net.