Moore made a nice sliding catch on an 18-yard throw from Drew Willy with seven minutes remaining in the fourth quarter to put the Bombers (1-0) up for good.
Willy completed 22 of 25 pass attempts for 325 yards and three touchdowns.
Saskatchewan (0-1) backup Kevin Glenn replaced Darian Durant at quarterback after Durant suffered an injury with 37 seconds remaining in the second quarter.
Durant wasn't touched on the play — an incomplete pass attempt to Jamel Richardson — but landed awkwardly on his left foot after he released the pass.
Last year, the Riders' star quarterback tore a tendon in his right throwing elbow in Week 11 — also against Winnipeg — and missed the remainder of the regular season and the West division semifinal.
Durant completed 13-of-18 pass attempts for 165 yards and two touchdowns before the injury. He connected with Taj Smith and Ryan Smith for the scores.
Durant was seen leaving the stadium on crutches midway through the fourth quarter with a protective boot on his left foot.
Jerome Messam took a handoff from Glenn and rumbled 53 yards for a go-ahead touchdown that gave the Riders a 20-15 lead to start the second half. Chris Getzlaf then caught a two-point convert from Glenn.
But the Bombers responded with a six-play, 75-yard drive that was capped with a 14-yard touchdown reception by Cameron Marshall. Marshall also scored on a two-point convert that tied the game at 23-23.
After a Chris Milo field goal, the Bombers again answered. This time, it was Willy to Moore that finished off another six-play, 75-yard scoring drive.
The Bombers' win was their first in Regina since 2004, carrying an 11-game losing streak into the matchup.
Cameron Marshall and Paris Cotton also scored touchdowns for Winnipeg.
The Roughriders lost middle linebacker Shea Emry late in the first half. Emry bumped his head on an attempted tackle and didn't return in the second half. Rookie receiver Nick Demski (undisclosed injury) also missed most of the fourth quarter.
Winnipeg slotback Clarence Denmark recorded his 250th career reception — an eight-yard catch — in the first quarter. He was helped off the field midway through the fourth quarter after a 41-yard reception and the Bombers trailing 26-23. Denmark is one of three receivers who reached the 1,000-yard receiving mark in 2014.