United registered their first league victory of the campaign as they dug deep to grind out an important result at the end of new manager Keith Curle's first week in charge.
Both sides started brightly without quite managing to find a way through as defences stood strong.
A purposeful run from Potts created space early in the day and he spotted Brough on the overlap on the left. The cross came early but Potts couldn't wrap his boot round the fizzed delivery and Tranmere hooked it clear.
United created a great opening with ten minutes on the clock when Amoo raced onto a pass from Dicker. He beat his man and popped a great ball into the box but Beck's downward header was put behind by the attentive Danny Holmes.
Grainger took charge of the corner kick and a winding run from Beck almost paid dividends when he rose highest. His header had power but it was always going wide.
Carlisle looked lively and White advanced to support another good move when Amoo called for help. The right back sent a beauty of a cross to the far post and Fon Williams had to spread himself to make a block on the line. Dicker was on to the half clearance quickly and Potts opened himself up to sweep the ball into the box. The Rovers keeper was on his toes and he did well to close the gap as Beck tried to finish it off.
Hanford did well to protect his lines as Tranmere got forward in numbers but it was all about United as they played with confidence and determination, particularly going forward.
The home side thought they'd taken the lead on 20 minutes when beck threw himself at a cross from the right. His touch left Dicker with time and he wheeled away in disappointment as his thundering toe poke skipped up and behind off the top of the bar.
A long throw from Grainger caused yet more problems for the visitors when Gill misjudged the flight of the ball. Amoo shielded it with his back to goal but the defender redeemed himself when he stepped across to hook it away.
The good Carlisle work was almost undone on the half hour when Bell Baggie linked with Richards inside the box. The striker showed great awareness to hold onto the ball as he danced past two challenges to go one-on-one with Hanford. The United stopper earned a round of applause as he stepped out to make a fantastic save.
Another long ball put Donnelly up against Thirlwell and the striker opted to shoot early as they reached the edge of the box. Hanford was untroubled as it bobbled towards the corner flag.
Carlisle had to take care just before the break when Kirby made it to the byline. He sent it into the danger area but Bell Baggie was on his heels and Grainger was able to shepherd it away.
Tranmere fired the first volley of the second half when Power and Kirby exchanged passes. There was a brief sight of goal but Power was off balance as he popped a weak effort well wide.
Donacien then took the bull by the horns when he ran out of his own half at the heart of the United defence. He found Donnelly but the shot was poor and Hanford gathered it in.
Tranmere created a chance from nothing on 57 minutes when Donacien robbed Beck deep inside the Carlisle half. The cross eventually fell for Power and it needed a half block from Thirlwell to send Power's volley over the bar.
Patience was needed as chances became few and far between but Carlisle got their noses in front when David Amoo bobbed and weaved his way onto a corner delivery. His header had Fon Williams flapping at thin air as it dropped over the line and into the back of the goal.
Rovers were handed a gilt-edged opportunity to level the scores when Bell Baggie and Kirby picked a path to put the ball on a plate for Donnelly who was stationed in a good position on the penalty spot. He snatched at it and was immediately substituted as his woeful effort flew high into the air.
United then had a chance to extend the lead when Amoo was sent clear by a pass from Paynter. He crossed early but Beck chose to go for the dummy as the crowd waited expectantly for a shot on goal.
Carlisle continued to dictate the play as they kept the ball high up the field and, in the end, they thoroughly deserved their first win of what has been a difficult start to the campaign.
United - Amoo (72)
United - Grainger (73)
Rovers - Bell Baggie (69)
Carlisle United - Hanford, White, Thirlwell, Archibald-Henville, Grainger, Amoo (Kearns (83), Potts, Dicker (Paynter 70), Brough (Elliott 77), Beck, Dempsey. Subs - Caig, Marrow, Rigg, Symington.
Tranmere Rovers - Fon Williams, Holmes, Power (Koumas 79), Ihiekwe, Donnelly (Stockton 77), Richards, Bell Baggie, Kirby (Fanimo 79), Donacien, Gill, Davies. Subs - Brezovan, Odejayi, Rowe, Laird.
Referee - Mr Jeremy Simpson
Attendance - 3,873 (240 away fans)