The Brewers dumped out Wigan and QPR last year and they added Neil Lennon’s men to their list of scalps thanks to Palmer’s deflected effort, which wrong-footed Ben Amos in the dying stages.
Wanderers would have had a half-time lead had Gary Madine converted the simple headed chance he was presented with when Remi Matthews’ save from Liam Feeney’s free-kick looped up to him.
However, from just four yards out, the ex-Sheffield Wednesday striker, who was denied a first Bolton goal at the weekend by the woodwork, planted his try on to the top of the crossbar, with the ball kissing the bar again before being cleared out.
Another pair of Madine headers, both off target, sandwiched Lawrie Wilson’s drive at Matthews as the hosts ended the first period in the ascendancy.
Aside from a deflected Lucas Akins try over the top, Burton did not threaten until Ben Amos rushed out to repel Crystal Palace loanee Jerome Binnom-Williams early in the second period.
Lennon’s men dominated for long second-half spells but Madine scooped well over the top following a mazy Mark Davies run, Neil Danns’ free-kick was pushed out and Zach Clough’s wayward effort hit the corner flag.
John Mousinho drove straight at Amos, and Feeney and Dorian Dervite did likewise when up against Matthews at the other end, before the Brewers finally found the net with three minutes remaining.
Palmer fired an effort at the Wanderers goal from the edge of the box and a big deflection caused the ball to divert the shot beyond Amos as he lay stricken on the ground.