Rogers pounced in the 75th minute, turning in a near post cross from substitute Sam Aiston, to deny Mansfield their first home win in three games.
Rodgers' strike cancelled out Mansfield's fortunate breakthrough, when Alex Neil's 64th minute deflected shot from 20 yards rewarded the home side for an improved start to the second half.
Mansfield had almost got off to a dream start when Adam Murray half-volleyed narrowly wide from 50 yards after 90 seconds with keeper Scott Howie stranded.
But the home side were lucky ten minutes later when Jake Sedgemore curled a 20-yard free-kick against the post.
When Kelvin Langmead's clever turn and shot was blocked by Alex Baptiste's last-ditch tackle it was the only other piece of goalmouth action in a scrappy first 40 minutes.
The game finally sprang to life four minutes before half time when another Sedgemore free-kick was knocked down by Langmead and Rodgers volleyed against the underside of the bar.
Mansfield finally tested Howie minutes later, but Richie Barker's shot after a neat Murray lay-off lacked power.
The Stags improved immediately after half time and an Adam Rundle cross-shot was finger-tipped over the bar by Howie.
At the other end Kevin Pilkington saved from Langmead and from his clearance Mansfield grabbed a lucky lead.
Neil cut in from the right by-line and his tame shot was deflected past a stranded Howie to give the midfielder only his second goal for the club.
Debutant Andrew Barrowman and Gareth Jelleyman were both denied a second goal by smart Howie saves.
Shrewsbury almost made Mansfield pay for missing those chances when Ryan Lowe broke clear, but with just the keeper to beat he blazed over.
However, moments later Rodgers levelled to stretch in-form Shrewsbury's good run to one defeat in seven games.
And in the 90th minute the visitors almost snatched victory but again Langmead failed to test the keeper when well placed inside the area.