A lack-lustre contest developed into a thriller after a stunning 25-yard strike by Burton midfielder Andy Corbett in the 34th minute. Rookie striker Liam Marum fluked an equaliser for Cambridge a minute after the interval.
But Quinn's hopes of a share of the spoils were dashed when Daryl Clare struck with his fourth goal of the campaign as Burton extended their unbeaten run to five games.
Albion boss Nigel Clough switched to 3-5-2 formation for the first time this season, with Aaron Webster being pulled out of the regular back four to make up a five-man midfield.
There was also a recall for Lee Fowler in place of Chris Hall, while Steve Scoffham dropped to the bench - despite his winning goal at St Albans - to make way for top scorer Clare.
Quinn was missing suspended midfielder Richie Hanlon and injured striker Robbie Simpson.
Albion produced a crisp passing move in the fourth minute, but Danny Holmes' shot was blocked on the edge of the area.
It was otherwise a tame opening until Clare's lovely cross-field pass in the 13th minute gave Corbett the opportunity to cut inside for a left-foot shot that also failed to find its way through a packed U's defence.
Marum sparked the visitors' attack into life in the 17th minute but, after doing the hard work by evading two tackles, his final pass was too strong for fellow striker Marcus Richardson.
Burton keeper Kevin Poole then did well to get his fingertips to dangerous cross from Courtney Pitt, but defenders had been very much in control of the opening 20 minutes.
Corbett's failure to capitalise on a stray pass by David Bridges was typical of the lack of fluency being shown by either team at this stage.
Interruptions for injuries to Burton striker Jon Shaw and U's full back Rob Gier merely added to the disjointed nature of the proceedings.
Burton flickered into life on the half-hour mark as they broke swiftly from a Cambridge corner kick only for Fowler to fire too high from 22 yards after a slick exchange of passes with Holmes.
The goal the game needed came in spectacular fashion in the 34th minute. Burton seemed to have squandered a promising move with a loose cross from Fowler, but the ball broke loose for Corbett to lash home a stunning left-foot shot from 25 yards.
Clare immediately threatened a second, squirming away from his marker to get in a scooped shot that Paul Crichton saved to his right.
There was now a brisk pace about the contest, and the U's could have been level in the 40th minute.
Trevor Robinson's free-kick from the left broke lose for Richardson, but the striker could only clip the top of the crossbar with a 15-yard drive.
The U's had briefly wrested the initiative away from the Brewers, though Burton had appeals for a penalty rejected after Clare's shot appeared to be blocked by the hands of Mark Peters.
Right at the end of the half, Ryan Austin produced a superb tackle to deny Marum a shooting opportunity after the striker had broken away from a mix-up in the Burton midfield.
The U's hit back with a fluke equaliser a minute after the restart. Marum received Robinson's pass close to the byline and could hardly believe his eyes when his firmly driven cross went in off the inside of Poole's far post.
Albion had reverted to a 4-4-2 formation, with Chris Hall on for Holmes, but it was the visitors who had their tails up and were denied the lead when Peters was penalised for pushing as he headed home a Jon Brady corner.
But the home side bounced back to regain the lead through Clare after an hour. Gary Rowett got his head to a Corbett cross and the Burton hit-man turned superbly on the edge of the area to rifle a right-foot shot into the bottom corner.
Clare fizzed a right-foot shot inches wide as the goal gave Albion the lift they needed after rocky spell.
It had developed into a pulsating encounter as the teams traded blows, Poole diving full-length to turn away Brady's 30-yard shot and Crichton was almost embarrassed by an audacious 40-yard effort by Clare that dropped a foot wide.