The recalled Simeon Jackson started up front in place of Amadou Bakayoko, who dropped to the bench, along with Maz Kouhyar and Liam Kinsella.
Rovers, above Walsall only on goal difference at the start of play, threatened first through Ellis Harrison, who headed straight at Neil Etheridge from a Boden cross.
The Saddlers – searching for their first away goal of the season – soon found their stride, with Jackson testing Pirates keeper Kelle Roos with a low 20-yard strike.
And Walsall deservedly broke the deadlock on 16 minutes, when the outstanding Oztumer curled a delightful left-footed shot into the bottom corner for his third goal in Saddlers colours.
The visitors were well on top and they should have been awarded a penalty moments later.
Oztumer’s incisive pass played in Jackson, who was brought down by Peter Hartley inside the area, but the referee - inexplicably - waved play on.
Six minutes after being denied a clear penalty at one end, the Saddlers conceded one at the other.
Captain Adam Chambers tripped Boden on the edge of the box, and the prolific Taylor – Rovers’ 28-goal top scorer last season – hammered the resulting spot-kick straight down the middle.
Darrell Clarke’s hosts almost snatched the lead just before the break, but Etheridge made a crucial, goal-line stop to claw away Taylor’s stooping header from a corner.
Walsall responded immediately, Oztumer with a wonderful reverse pass through to Jackson but his effort was well blocked by the advancing Roos.
Half-time: Bristol Rovers 1-1 Walsall
Rovers had a big chance to go in front on 53 minutes, when Brown’s cross was headed back across goal by Taylor and flicked onto the top of the bar by Harrison.
The Gas began to increase the pressure, with Harrison heading over from Daniel Leadbitter’s right-wing cross and Bodin having a goalbound shot blocked by Joe Edwards.
Just before the hour-mark, Etheridge was forced into a fine double save; he first parried Taylor’s fierce drive, before getting up quickly to push away Bodin’s follow-up.
Walsall were less threatening in the second half but had a great opportunity to restore their lead when Morris was put clean through by Jackson but he ballooned his effort over the bar.
The Pirates went back on the offensive, with Etheridge – a former Rovers loanee – making a spectacular full-stretch stop to turn Bodin’s curling strike wide.
The home side looked the more likely to find a winner.
A left-wing corner flashed across the face of goal, just eluding the unmarked Harrison, while a Lines free-kick was palmed wide by Etheridge.
In stoppage time, Taylor saw a low drive deflect narrowly wide of the near post, and Rory Gaffney dragged an effort off target after latching onto Taylor’s back-heel.
Full-time: Bristol Rovers 1-1 Walsall
Team: Etheridge, Edwards, McCarthy, Toner, Laird, Chambers, Moussa (Bakayoko 84’), Morris, Makris (Kouhyar 36’), Oztumer, Jackson.
Subs not used: MacGillivray, Dobson, Kinsella, Preston, Sangha.
Attendance: 7,805 (170 Walsall supporters).