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.
Sheffield United's poor form going into the League One play-offs continued as they were beaten 1-0 at Crewe Alexandra .
The Blades had already booked their play-off place but Aneke's winner means they have won just one of their last seven games.
Harry Maguire almost gave the visitors an early lead but his header was saved by Steve Phillips and Kelvin Mellor did well to deflect Neil Collins' follow-up over the bar.
Jonathan Forte sent a couple of efforts from the edge of the box high and wide before Phillips was in action again.
The keeper dived low to parry Callum McFadzean's drive and then recovered quickly to smother Chris Porter's rebound.
Crewe's Oliver Turton and Harry Davis sent efforts off target while Aneke's daisy-cutter was tipped around the post by George Long as the home side struggled to create any clear-cut first-half chances.
But Aneke stunned the visitors four minutes after the break when he beat Long with a superb 25-yard finish to put the Alex in the driving seat.
A fierce Barry Robson free-kick which Phillips did well to palm away almost got the Blades back in it while the keeper got a fingertip to a late shot from Joe Ironside.
At the other end, Aneke looked to double his account with a volley which dipped over the bar and a low effort which he dragged wide after a surging run.
An early goal from midfielder Mike Jones gave Crawley Town a deserved 1-0 victory and condemned Preston North End to their first defeat in five games.
Jones drove Crawley ahead after only six minutes with a bouncing right-foot shot from 25 yards into the bottom corner.
Skipper Josh Simpson shot over from the edge of the area and keeper Thorsten Stuckmann denied Paul Hayes as the hosts dominated.
Stuckmann then did superbly to tip over a goalbound header from striker Billy Clarke after Hayes' neat assist.
Preston's first attempt on goal did not arrive until the 33rd minute when captain John Welsh put a 30-yard free-kick wide.
Crawley ended the half totally on top and Clarke's cleverly-played free-kick was saved by Stuckmann.
Home keeper Paul Jones denied Jack King in a one-on-one situation shortly after the break and on-loan striker Hayes was similarly thwarted at the other end by Stuckmann.
Preston, who improved after introducing three substitutes in the second half, came close to snatching a point 20 minutes from time when King headed a Lee Holmes cross against the crossbar.
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