City threatened first on 12 minutes when Steven Howard's flick-on found Matty Fryatt whose powerful angled drive was tipped around the far post by the Scunthorpe keeper Joe Murphy.
The Foxes went ahead after 16 minutes when Howard's header from a Max Gradel corner was blocked by a defender with the ball falling to Morrison who fired home from five yards.
As the first half drew to a close Gradel tested Murphy with a dipping free-kick before Scunthorpe had a narrow escape when Lloyd Dyer struck the bar after David Mirfin had cleared Fryatt's shot off the line.
United were level on 57 minutes when Martyn Woolford was allowed to get in a low cross from the right which Paul Hayes slotted home from seven yards.
Scunthorpe found themselves in front on 66 minutes when substitute Bruno Berner was ruled to have hauled down Hayes inside the box with Grant McCann confidently stroking home the penalty with David Stockdale diving the wring way.
Leicester leveled on 76 minutes when Jack Hobbs nodded a Matty Oakley free-kick across the face of goal for Morrison to head home from five yards.
The result leaves Scunthorpe with a tense final two games as they moved ahead of Tranmere on goal difference into the final play-off place.