Icelandic international Bjornsson was sharpest to react to Leon McSweeney's cross from the left in the fourth minute nipping in between a defender and keeper Alan Collin to steer the ball home from six yards.
Harte finally got Carlisle back on level terms in the 33rd minute, squeezing the ball home from just a couple of yards. Taiwo then grabbed his first ever League goal in the 52nd minute to give the Cumbrians the lead.
Former Pools winger Robson was responsible for Carlisle's third goal in the 63rd minute when his speculative cross from wide on the left deceived Flinders and the goalkeeper was left red-faced as the ball hit the underside of the crossbar then rebounded into the net off his body.
Monkhouse gave Pools some hope in the 90th minute when he swept home a left-wing cross but it was not enough to prevent Hartlepool going down to a third straight defeat.