Jacobson, the stand-in captain, chose a perfect time to grab his first goal of the season when he whipped a 20-yard effort over the Oldham wall and high into the net four minutes into the second half.
Both of these sides guaranteed League One survival last weekend and it was Shrewsbury who started the brighter.
They should really have forged an early lead but Akwasi Asante, having pounced on a slip from Jean Yves M'Voto, fired wide from the edge of the area with only goalkeeper Alex Cisak to beat.
Oldham, whose manager Lee Johnson made eight changes to the team which beat Crawley last Saturday, then created a stream of chances.
Top scorer Jose Baxter twice stung the gloves of goalkeeper Chris Weale, who also saved sharply from Kirk Millar.
Then Baxter squandered a golden chance right on half-time when he headed Lee Croft's cross wide.
But Shrewsbury grew increasingly dominant in the second half and might have added to their lead when Jon Taylor rounded Cisak but fired against the near post from a tight angle.
Then Connor Goldson headed a corner just off target but Jacobson's fine strike proved enough to lift Shrewsbury above Oldham in the lower reaches of the table.