The Hammers pair, who have both featured in all three of the side’s Championship matches so far this campaign, could not help Phil Parkinson’s men to a point against the Rams.
Gary Rowett’s visiting men raced into an early lead when Dave Nugent scored from close range just eight minutes in.
And things went from bad to worse for the hosts less than a quarter of an hour later when the former England international grabbed his second of the afternoon at the Macron Stadium, heading home Johnny Russell’s cross.
Bolton steadied the ship from then on in, and did halve the deficit in injury time when Gary Madine pounced, but it was too little too late for Cullen and Burke’s side who slipped into the bottom three in England’s second tier after the defeat.
Bolton host Sheffield Wednesday on Tuesday in the Carabao Cup second round.
Meanwhile, Reece Oxford recovered from an ankle injury in time to be named on the bench in Borussia Monchengladbach’s opening Bundesliga match and derby win over FC Koln on Sunday.
The hosts were dominant throughout and scored the game’s only goal four minutes after half time, when right-back Nico Elvedi bagged his first in Borussia colours, converting Ibrahima Traore’s cross.
Oxford’s men are next in action on Saturday when they travel to Augsburg in the Bundesliga.