Grounds pounced after 13 minutes when he slotted in from six yards out after goalkeeper Chris Weale could only palm out Matt Derbyshire's low cross.
Weale then saved from Matt Smith before Shrewsbury captain Matt Richards hit a long-range free-kick to test home keeper Dean Bouzanis.
Mark Wright was also denied by Bouzanis but Oldham wasted chances as Smith sent a header and a half-volley wide of the target.
The Shrews could have equalised on the hour mark, only for Luke Rodgers to see his goalbound effort blocked by Jean Yves Mvoto.
Luke Summerfield fired a free-kick just over for the visitors, who remain without an away win this season.