Melbourne were also unlucky not to score in the 20th minute through Mike Harris, who looked certain to touch down but for some desperate defence by Stormers flanker Schalk Burger, who held him up over the line.
Two minutes after that another Rebels error led to the home side's opening try.
Rebels winger Tom English won the race to a Stormers' kick down field but flicked it to their halfback Louis Schreuder, with the ball then finding centre Damien De Allende to score.
Leyds got his second try of the match in the 45th minute when after a big Stormers scrum shove they sent the ball wide before the winger cut back inside.
With sharp-shooting five-eighth Demetri Catrakilis adding the extras the score was out to 28-10.
Melbourne continued to toil hard and were rewarded with a try through English while flanker Sean McMahon almost got across in the 76th minute but the margin was too great to peg back.
Rebels skipper Scott Higginbotham lamented their high error count and said a video review of the match would be tough to watch.
"It's going to be a tough one to review because our errors would have been through the roof and that's disappointing," he said.
"We played some great football but credit to the Stormers, they really kept it in play for 80 minutes."
-AAP
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