Carlisle were rarely troubled by the visiting attack and Tranmere's cause was not helped having to play with 10 men from just before half-time after defender Shaleum Logan received his marching orders for a second yellow card in the space of 30 seconds, both for fouls on Matty Robson.
Harte was Carlisle's first-half danger man, forcing an early fumble from keeper Luke Daniels with a raking 35-yarder. The former Leeds man went even closer in the 42nd minute with a 30-yard free-kick blast which bounced off the crossbar with Daniels well beaten.
Carlisle were totally dominant after half-time and only dogged defending and some fine saves from Daniels kept Tranmere level until the 59th minute when Robson sneaked in behind the defence and his cross was turned in from six yards by Anyinsah.
Carlisle went further ahead in the 68th minute when skipper Livesey got on the end of Kevan Hurst's curling cross to head home from the tightest of angles. And the third goal followed in the 77th minute when Harte crashed home a near-post header from Robson's corner on the left.