Home goalkeeper Cameron Belford did his best to keep the Blues in the game and give them the chance to progress but the Fir Park men finally got the better of him in the second half and booked their place in the next round.
The first half ended goalless, although it was not for the want of trying by either side, with Lionel Ainsworth setting the tone on four minutes from 25 yards that forced Belford into a fine stop. Scott Agnew tried his luck from a similar distance for the hosts six minutes later but his effort flashed wide.
Midway through the half the Fir Park side created the best opening of the opening 45 minutes when Ainsworth picked out Scott McDonald, pictured. The striker shook off his marker before shooting but Belford kept his goal intact.
The hosts appealed in vain for a penalty shortly afterwards when Mark McGuigan claimed that he was pushed in the back as he stretched for a Scott Robertson cross but nothing was given.
Belford ensured that both sides drew a blank before the break when he used a foot to keep out an Ainsworth effort.
A goal finally arrived six minutes after the break with Cadden cutting in from the left wing and sending a powerful effort past the previously unbeatable Belford from the edge of the box.
Mark McGhee’s side almost added a second when a lightning breakaway gave Johnson an opening but his low effort was saved.
Craig Malcolm gave Motherwell a reminder that Stranraer were still in the game on 55 minutes when he fired for goal but his effort cleared the bar. Stephen McManus looked to seal Motherwell’s win with a shot from 40 yards but the ball just drifted wide of the home goal.
Cadden was denied another goal on 63 minutes when his floated effort to the back post was tipped over by the overworked Belford.
Motherwell finally wrapped up the win seven minutes from time when Cadden took advantage of hesitancy in the home defence and bore down on goal before teeing up Johnson to tap home.
With Stranraer realising that their challenge was over, McDonald nodded in a Johnson cross to end the scoring four minutes from time.