After a relatively even first half, it was Scunthorpe who took the lead through Paddy Madden's close-range header on the stroke of half-time - the forward's ninth goal of the campaign.
But the hosts had the better of the second period, and managed to grab an equaliser in the 69th minute when Sean Geddes drilled in a low shot into the bottom-right corner after latching onto Daniel Nti's pass.
Geddes went close again late on but the match proceeded into a scrappy extra-time period, with neither side able to force a breakthrough.
An enthralling penalty shoot-out followed, with a total of 32 spot-kicks being taken, but Carl Heeley's Worcester were left facing elimination when defender Wayne Thomas' effort was parried away by Scunthorpe goalkeeper Sam Slocombe.
Scunthorpe defender Miguel Llera, who had missed an earlier spot-kick, stepped up to atone for his error and send Mark Robins' men into the third round, where they will face either Chesterfield or MK Dons.