First half goals from Sweeney and Hope put United in the driving seat and they complemented that with a 90 minute performance of grit and purpose to comfortably book their progression into round two of this season's FA Cup competition.
United fashioned the first opening of the day when a clipped ball over the top found Asamoah on a charge. He set himself and was disappointed to see a crisp volley go wide.
A wayward shot from Brunt flew into the stands as he spun and let one fly with his back to goal, but it was the pace of Asamoah which caused most of the early problems as he covered every blade of grass up front.
He was quick to pounce on a loose pass from Tanner which gave him the chance to race into the box but McCormick was off his line quickly to close the angles. A toe poke beat the keeper but bobbled harmlessly past the upright.
Plymouth enjoyed a good spell without quite managing to open Carlisle up until a rushed back pass from Grainger sent Carey on the chase. Hanford moved smartly to make the block and clear the ball.
A rash challenge from Raynes on Carey brought him a fifth booking, and a pending one-match suspension, and the resulting free kick caught the visitors on the back foot as it bent to the back stick. Nelson got there with a downward header which beat both the keeper and the far post.
Carey looked to be up for the challenge and he tussled with Atkinson to take the ball into the United half. Simpson took over but a fizzed shot was blocked by Raynes and Hanford was able to gather with ease.
More movement from Asamoah took him onto a slide rule Gilliead pass, as the Cumbrians started to ask even more questions, but McCormick got an important touch to prevent the ball from getting anywhere near the net.
United kept the pressure on and a deep free kick from Grainger had the Pilgrims confused when a deft touch from Ellis sent it into the mix. A half clearance was manipulated by Raynes and Gilliead showed awareness to bring Antony Sweeney into the mix. He didn't disappoint with a fantastic scissor-kick volley on the spin which flew past Luke McCormick.
Confidence grew in the visiting ranks and Sweeney pressed to win the ball back as Jervis hesitated in the centre circle. Asamoah was onto the through ball but he was off balance as he let fly and it rose high over the bar.
Wide man Alex Gilliead then cut in from the right to ghost past two challenges and his one-two with Asamoah appeared to have done the job. A heavy touch allowed the keeper to pounce and pull it into his arms before the youngster could pull the trigger.
It was all about the blue shirts and a Grainger free kick evaded everyone as it skipped up off the surface. Ellis showed for it but it was a difficult chance and he couldn't keep it down.
The Cumbrians were reminded that there was defensive work to be done when a Tanner free kick bent wickedly as it reached the six-yard box. Raynes completed a superb covering header to somehow send it over his own bar.
Carlisle kept the pressure on despite the momentary lapse and a piece of midfield magic from Gary Dicker set Asamoah free yet again as a weighted ball over the top left Plymouth at sixes and sevens.
Excellent chest control from the forward opened the door but McCormick was equal to it as he dived low to palm the goal-bound strike away. The relief was short-lived though as Hallam Hope, granted FA Cup clearance from parent club Bury just 24 hours earlier, pounced to bundle the spinning ball past the defender and into the gaping net.
Plymouth enjoyed tentative pressure after the break but their attempts to find a way through fell against a solid blue wall with every man working hard to keep the shape and maintain the discipline. Tanner and Carey sent deliveries into the box but Ellis and Raynes kept their concentration to complete a number of important clearances.
The Cumbrians picked their moments to get forward and a slip from Simpson gave Asamoah half a sight. He pushed forward but he could only find the side netting as two men converged to make a block.
Tanner looked the most likely for the Pilgrims but two shots in the space of three minutes went much the same way - off target and well wide.
Carlisle sat deeper to protect their lead and they had to little to worry about as long balls forward were easily dealt with. The final whistle brought a standing ovation from the 176-strong travelling contingent as they acknowledged what could arguably have been the best performance of the season so far.
United - Sweeney (23), Hope (41)
Plymouth - Mellor (22), Carey (62), McHugh (77)
United - Raynes (15), Sweeney (72)
Plymouth Argyle - McCormick, Mellor, Sawyer, McHugh, Nelson, Hartley, Simpson (Wylde 72), Carey, Jervis, Brunt (Harvey 84), Tanner (Threlkeld 84). Subs - Bittner, Cox, Purrington, Rooney.
Carlisle United - Hanford, Atkinson, Ellis, Raynes, Grainger (Miller 78), Gilliead (Thompson 84), Dicker, Sweeney (Hery 74), Kennedy, Hope, Asamoah. Subs - Gillespie, Wyke, Archibald-Henville, McQueen.
Referee - Mr Charles Breakspear
Attendance - 6,005 (176 away fans)