Aston Villa loanee Kevin Toner was handed his debut at centre-half, while Liam Kinsella was brought in at right-back.
James O’Connor and Franck Moussa both missed out through injury, with George Dobson and Matt Preston dropping to the bench.
The Saddlers entered the match searching for their first home win against Bury in 20 years, and they made the perfect start, taking the lead inside six minutes.
Oztumer’s intelligent reverse pass played Makris in behind the Bury defence and the Cyprus international accurately cut the ball back for Morris to fire high into the roof of the net.
Bury should have drawn level immediately as Vaughan headed a whisker wide from Zeli Ismail’s right-wing cross when well-placed.
The Saddlers, though, had started with a real purpose and on 14 minutes they doubled their advantage.
After Barnett had conceded possession sloppily midway inside his own half, Oztumer advanced forward and steered an unerring left-footed drive beyond Ben Williams from 25 yards.
David Flitcroft’s visitors looked to reduce the deficit, with Mayor shooting straight at Neil Etheridge, who then kept out a firm Hallam Hope strike.
But Walsall added a third goal just past the half-hour mark, as Jackson nipped in ahead of Williams to turn Makris’ cross home from close range.
And the home side really should have made it 4-0 moments later, when a dreadful error from Shakers skipper Antony Kay allowed Jackson to run clean through on goal.
The Canadian striker elected to shoot early but Williams got down well to make a fine one-handed stop.
Jackson was presented with another one-one-one opportunity on the stroke of half-time, as he pounced on Greg Leigh’s misplaced pass, but his shot was blocked by Williams.
Half-time: Walsall 3-0 Bury
Walsall appeared in complete control, but the Shakers reduced the deficit just moments after the interval.
Vaughan’s cross was tipped over by Etheridge and from the resulting corner, Barnett hammered a powerful strike into the top corner to give Bury a glimmer of hope.
And barely two minutes later the Shakers were right back in it, as Vaughan drilled past Etheridge at his near post from Barnett’s knock-down.
The Saddlers were seriously under the cosh as Bury - with their confidence and belief now restored – went in search of an equaliser.
Vaughan lashed over on the turn and Tom Pope headed straight at Etheridge, while Kay and Mayor each saw attempts blocked.
And the pressure told on 66 minutes, as Jason McCarthy could only head Craig Jones’ dangerous cross as far as Mayor, who slammed home from 12 yards to bring Bury level.
With Walsall rocking, the visitors almost edged in front as Vaughan’s half-volley flashed a fraction over.
The Saddlers finally managed to settle down and could have won it late on, but Williams produced a fine flying save to palm away Morris’ 25-yard free-kick.
Full-time: Walsall 3-3 Bury
Etheridge, Edwards, McCarthy, Chambers, Jackson, Oztumer (Kouhyar 73’), Morris, Osbourne (Dobson 43’), Kinsella, Toner, Makris (Bakayoko 58’).