Andy Keogh capitalised on a mistake from David Edgar to open the scoring before Wright illuminated the first half and left Burnley staring at back-to-back defeats.
Substitutes Martin Paterson and Danny Ings almost brought Eddie Howe's side back into the contest as the hosts cranked up the pressure after half-time.
However, Harry Kane settled matters on the counter-attack with 70 minutes gone to move Millwall seven points clear of the relegation zone.
Jay Rodriguez pulled a goal back with an injury-time penalty, but a third defeat in four outings leaves manager Howe's side six points shy of the play-off places.
The visitors thought they had taken an early lead when Hamer Bouazza's effort from 18 yards cannoned off the underside of the bar and went close to crossing the line. As the referee's assistant declined to signal for a goal, Lions midfielder James Henry headed over from the rebound.
Keith Treacy fired narrowly wide from the edge of the area as Burnley tried to feel their way into the match, but when Kane capitalised on some hesitant play by Ben Mee to burst clear it took a desperate last-ditch interception from Marvin Bartley to prevent Millwall going ahead in the 12th minute.
From the resulting corner Bartley was the hero once more, clearing Paul Robinson's header off the goalline, but his efforts would soon prove to be in vain.
Clarets defender Edgar botched a routine clearance and Keogh was on hand to provide a clinical left-footed finish.
Charlie Austin narrowly failed to connect with a Treacy cross as the host sought an immediate response, and the former Swindon striker left referee David Phillips unmoved when he went down under a challenge from Bouazza in the area.
But Millwall doubled their advantage in spectacular fashion after half an hour. Henry's left-wing free-kick was headed clear by Mee and the ball looped up for Wright to thrash home an unstoppable 30-yard volley.
A miserable half was complete for Burnley when captain Chris McCann hobbled out of the action to be replaced by Paterson.
Howe elected to use his two remaining changes at the interval, with striker Ings and defender Brian Easton replacing Treacy and Austin.
The replacements almost had the desired effect, with Paterson inches away from converting a Wallace free-kick in the 53rd minute and Ings reacting quickest on the edge of the area to fizz wide with an hour gone.
Ings and Wallace were agonisingly off target once again as Burnley ramped up the pressure, before Millwall sealed the points on the break with 20 minute to play.
Henry benefited from the extra space on offer in the Clarets half to find Kane, who swept past advancing goalkeeper Lee Grant.
Rodriguez reduced the arrears in time added on, netting his 19th of the season after Shane Lowry fouled Bartley.
