The hosts went ahead in the 23rd minute, Amadou Bakayoko drilling his first league goal past Jayson Leutwiler from Oztumer's fine through ball.
Shrewsbury levelled on the half hour as former Walsall player Oliver Lancashire looped home a header from Ian Black's free-kick.
The Saddlers regained the lead three minutes before half-time, Kieron Morris bundling home Joe Edwards' cross from inside the six-yard box.
Oztumer should have sealed the points 10 minutes into the second half, but fired wide when one-on-one with Leutwiler.
And the Shrews levelled in the 58th minute as Walsall goalkeeper Neil Etheridge missed his punch from a corner and Junior Brown bundled home from close range.
Shrewsbury were then denied a goal and a penalty by the linesman as both an Andy Mangan strike and a foul on George Waring in the box by James O'Connor were cancelled out by offside flags.
The visitors went down to 10 men after 65 minutes as Ryan McGivern received a second yellow card for an off-the-ball clash with Bakayoko.
And Walsall won it two minutes from time as Oztumer's corner crept in direct at the near post, although they also ended with 10 men as Bakayoko saw a second yellow for his celebrations.
Reports supplied by the Press Association.
Match ends, Walsall 3, Shrewsbury Town 2.
Second Half ends, Walsall 3, Shrewsbury Town 2.
Attempt missed. Jim O'Brien (Shrewsbury Town) right footed shot from outside the box is just a bit too high.
Substitution, Walsall. Matt Preston replaces Franck Moussa.
Substitution, Walsall. Andreas Makris replaces Erhun Oztumer.
Second yellow card to Amadou Bakayoko (Walsall) for excessive celebration.
Goal!  Walsall 3, Shrewsbury Town 2. Erhun Oztumer (Walsall) left footed shot from a difficult angle and long range on the right to the bottom right corner.
Corner,  Walsall. Conceded by Oliver Lancashire.
Attempt missed. Franck Moussa (Walsall) right footed shot from the centre of the box is close, but misses to the left.
Corner,  Walsall. Conceded by Junior Brown.
Delay over. They are ready to continue.
Substitution, Shrewsbury Town. Sylvan Ebanks-Blake replaces George Waring because of an injury.
Attempt saved. Franck Moussa (Walsall) right footed shot from the left side of the box is saved in the top centre of the goal.
Attempt blocked. Franck Moussa (Walsall) right footed shot from the centre of the box is blocked.
Attempt blocked. Antoni Sarcevic (Shrewsbury Town) right footed shot from the centre of the box is blocked.
Attempt saved. George Waring (Shrewsbury Town) header from the centre of the box is saved in the centre of the goal.
George Waring (Shrewsbury Town) wins a free kick in the attacking half.
Foul by Kevin Toner (Walsall).
Substitution, Walsall. Josh Ginnelly replaces Kieron Morris.
Jack Grimmer (Shrewsbury Town) wins a free kick in the attacking half.
Foul by Joe Edwards (Walsall).
Ian Black (Shrewsbury Town) is shown the yellow card for a bad foul.
Foul by Ian Black (Shrewsbury Town).
Adam Chambers (Walsall) wins a free kick in the defensive half.
Corner,  Walsall. Conceded by Jayson Leutwiler.
Attempt saved. Erhun Oztumer (Walsall) left footed shot from outside the box is saved in the bottom left corner.
Substitution, Shrewsbury Town. Jim O'Brien replaces Andy Mangan.
Foul by George Waring (Shrewsbury Town).
(Walsall) wins a free kick in the defensive half.
James O'Connor (Walsall) is shown the yellow card for a bad foul.
George Waring (Shrewsbury Town) wins a free kick on the left wing.
Foul by James O'Connor (Walsall).
Attempt blocked. Abu Ogogo (Shrewsbury Town) right footed shot from the centre of the box is blocked.
Amadou Bakayoko (Walsall) is shown the yellow card.
Second yellow card to Ryan McGivern (Shrewsbury Town).
Attempt blocked. Joe Riley (Shrewsbury Town) right footed shot from outside the box is blocked.
Ian Black (Shrewsbury Town) wins a free kick in the attacking half.
Foul by Adam Chambers (Walsall).
Goal!  Walsall 2, Shrewsbury Town 2. Junior Brown (Shrewsbury Town) left footed shot from very close range to the centre of the goal following a corner.
Corner,  Shrewsbury Town. Conceded by Joe Edwards.
Walsall piled the misery on managerless Shrewsbury as Erhun Oztumer's late winner settled a bad-tempered game in the Saddlers' favour.