Ian Thomas-Moore opened the scoring from the penalty spot early on on Friday night and added a second from 12 yards following a foul by Fraser Forster which saw the Canaries goalkeeper dismissed by referee Eddie Ilderton. Craig Curran added a controversial third for Rovers shortly after the half-hour amid strong appeals for handball and, although Grant Holt pulled a goal back, Paul Lambert's misery was compounded when he was sent to the stands in the second half.
Thomas-Moore's first penalty came after Russell Martin had been penalised for handball. The striker made no mistake with his spot-kick and confidently slotted a second penalty past substitute goalkeeper Declan Rudd after being brought down by Forster, who received a straight red.
Things got worse for the visitors when Curran fired home from 12 yards after appearing to control the ball with his arm.
Despite their numerical disadvantage, the Canaries did create a number of chances and eventually got on the scoresheet in the 58th minute as Holt converted Martin's cross.
Norwich boss Lambert was then dismissed after running onto the pitch to kick the ball to Rudd following an offside decision and Tranmere almost added a fourth with seven minutes remaining when Thomas-Moore struck the woodwork.