Kamara's first Premiership goal for Albion and Campbell's first goal of the season earned Albion their second win in four games.
City's leading scorer Andy Cole, who was making the 400th league start of his career, capped a dismal day for Stuart Pearce's side when he was dismissed with four minutes remaining.
Cole, who had also been sent-off on his previous visit to The Hawthorns with Fulham last season, saw red after collecting two yellow cards in the space of two minutes for a couple of rash challenges.
Senegal international Kamara, who joined Albion in the summer from the Italian side Modena, pounced after just five minutes with Campbell, who had scored his first goal for Albion in the corresponding fixture last season, sealing the win in the 61st minute.
City looked a pale shadow of the side that had hammered Charlton 5-2 at The Valley last weekend to secure their first away win since August.
They never recovered from Kamara's early strike as Albion ended the unwanted statistic of never having won a Premiership game in December.
Kamara could have broken the deadlock 60 seconds before he did pounce as only a smart save from David James prevented him scoring with a snap-shot.
But England goalkeeper James was powerless to prevent Kamara striking in the fifth minute after a raking cross-field ball from Junichi Inamoto.
The Japan international cut in from the left wing and drilled the ball into the path of Kamara.
Kamara was faced with the City left-back Ben Thatcher but easily cut inside the former Wales international and drilled a right-foot shot past the outstretched right hand of James.
It was a setback that City struggled to get to grips with and they were fortunate to reach half-time without suffering any more damage.
Albion's leading scorer Nathan Ellington twice went close before turning provider for Kamara who was denied by James surging out of his goal to tip the ball away from the former Portsmouth winger.
Having spent so much of the opening period on the defensive City struggled to test Albion goalkeeper Tomasz Kuszczak.
The closest City came to threatening before the break was on the half hour when Trevor Sinclair lifted the ball over the bar after being picked out by a right-wing centre from David Sommeil.
City did at least manage to start the second period in a more positive frame of mind and they should really have been on level terms in the 49th minute.
When Inamoto surrendered possession to Sinclair his mis-hit shot found its way into the path of Cole. Cole was only faced with Kuszczak to beat but the former England international fired his volley over the bar.
It was a costly miss as Albion put the game out of City's reach with Campbell's clinical 61st minute header. While his finish from eight yards was precise, the goal owed everything to the persistence of full-back Paul Robinson who wrapped his foot around the ball to deliver the centre when the pass from Kanu had looked set to roll out for a goal kick.
MAN OF THE MATCH: Paul Robinson (West Brom) - The left-back was tenacious in the tackle and led by example. He also set up Albion's second goal to add growing weight to claims that he should be in contention for an England call-up.