Newport's top scorer tapped in Alex Rodman's set-up as they moved 10 points clear of the League Two drop zone with their first home win since November.
County dominated and Boden had wasted a first-half one-on-one when the striker dragged his shot wide.
Carlisle ended the game with 10 men as Brandon Comley was sent off for a second yellow late on.
Newport's only concern was an injury to defender Scott Barrow, who was taken off on a stretcher with a head injury.
Scott Boden eased Newport County's relegation worries with his 11th league goal of the season to defeat Carlisle.