Omar Daley set the ball rolling with the opener and Jamie Murphy claimed a double before Ojamaa put the gloss on the win late on.
The Irn-Bru Third Division side pulled off a major shock when they dumped Aberdeen out of the League Cup in 2006 but they were unable to claim a second Scottish Premier League scalp at Fir Park on Saturday.
Motherwell's new signing Ojamaa started on the bench after joining the Steelmen on a short-term deal until the end of the season.
But the opening spell was all about Daley, who tested goalkeeper Neil Parry with a stinging shot in the first minute.
Queen's had a go themselves when Martin McBride rifled narrowly over the bar but they found themselves trailing after just six minutes.
Daley cut inside from the right and did well to hold off two defenders to unleash an angled drive that found the back of the net via the inside of the post.
The visitors tried to haul themselves back into the tie but a header from David Murray was comfortably collected by goalkeeper Darren Randolph.
The task became even more difficult when Motherwell doubled their advantage after 21 minutes when Murphy left Parry with no chance with an opportune strike from 25 yards.
Only the quick reactions of the Queen's Park goalkeeper prevented them from falling further behind when he did well to block a one on one from Bob McHugh, who had skipped past James Brough with ease to set up the shot.
At the other end, Ian Watt dragged a well-struck effort across the face of goal and wide, before he was the victim of a crunching tackle that earned Ross Forbes the first booking of the game five minutes before the break.
To their credit, Queen's Park emerged for the second half determined to have a go and a long, low shot from Jamie Longworth forced a decent save from Randolph, who just turned the effort past the post.
Watt then whipped a shot inches past the upright as the Spiders enjoyed a positive period early in the second half.
However, Motherwell could have all but ended any hopes of a comeback when a ferocious shot from Forbes appeared netbound only for a full-stretch Parry to tip just over the bar.
Ojamaa was handed his debut when he was thrown into the action for McHugh after 67 minutes, with Forbes also making way for Chris Humphrey.
Queen's had battled bravely but never really looked like posing any major problems for Stuart McCall's men and the tie was effectively over as a contest with 71 minutes gone.
Well claimed their third of the day when Humphrey tried to set up Ojamaa for the shot and the ball slipped through his legs and fell for Murphy to lash home and secure himself a double.
Ojamaa then sealed the win in the final minute of the game when he collected a cut-back from Humphrey before firing past Parry as Motherwell eased into the next round.