But the Saddlers equalised 17 minutes from time, as Oztumer fired into the bottom corner after a neat one-two with Simeon Jackson.
Walsall are now unbeaten in six matches and have only lost one of their ten in all competitions.
Both teams entered the match in fine form, having each won four of their previous five matches in all competitions.
Kieron Morris had the game’s first attempt, firing straight at visiting keeper Lee Burge from outside the box.
Coventry, who had Mark Venus in caretaker charge, enjoyed plenty of possession in the first half and twice went close through Ruben Lameiras.
The Portuguese midfielder first curled inches wide from 20 yards after good work from Daniel Agyei, before having a deflected shot held by Neil Etheridge.
Walsall’s best opportunity of the opening period came on 24 minutes.
Jason McCarthy surged forward down the right and accurately pulled the ball back for Andreas Makris but, under pressure, the Cypriot international fired wide at the near post.
The breakthrough came eight minutes before half-time, when Marvin Sordell’s low right-wing cross was turned home by Rose from eight yards.
Walsall fashioned little in response, although captain Adam Chambers did sting the palms of Burge with a rasping 30-yard strike.
Half-time: Walsall 0-1 Coventry
City could have doubled their lead within 30 seconds of the resumption, as Agyei burst past James O’Connor into the area before blazing well wide when he should have done much better.
The visitors continued to enjoy the better of proceedings, with Agyei’s powerful right-footed drive forcing Etheridge into a smart reaction save at his near post.
Walsall had a huge chance to draw level midway through the second half when Joe Edwards’s left-footed shot was saved one-handed by Burge.
The loose ball ran for Oztumer, who went down under a defender’s challenge but no penalty was given, with Morris then drilling wide with the goal gaping.
But the Saddlers did manage to find an equaliser on 73 minutes, Oztumer finding the bottom corner with an accurate left-footed shot after a neat one-two with Simeon Jackson.
A team transformed, Walsall pushed for a winner with Morris’ fizzing drive pushed over by Burge, who then kept out Josh Ginnelly’s near-post volley.
The home side were in the ascendancy and Liam Kinsella almost won it in the closing stages as he fired just wide from Oztumer’s intelligent pass.
Full-time: Walsall 1-1 Coventry
Team: Etheridge, McCarthy, O’Connor, Toner, Laird, Chambers, Edwards, Morris (Kinsella 89’), Moussa (Ginnelly 72’), Oztumer, Makris (Jackson 62’).
Subs not used: MacGillivray, Dobson, Osbourne, Preston.
Attendance: 6,719 (1,981 Coventry supporters).