Striker Danny Whitehall's smart finish with the outside of his right foot fired the hosts ahead after 33 minutes.
Substitute Lyle Simpson was then on hand to add a second on 87 minutes to secure the Spitfires' passage into the quarter-finals.
Hawks, coming up against a largely youthful Easltleigh side, enjoyed plenty of possession - particularly in the second half - but were unable to trouble Ross Flitney in the hosts' goal on the whole as they exited the competition they have won in three of the past five seasons.
Unsurprisingly Hawks were much-changed from the weekend National League South defeat to Bath City. In total, there were seven changes with just captain Joe Oastler, Billy Clifford, Joe Newton and James Roberts in the starting line up at the Silverlake Stadium who also featured from the outset against Bath.
Lively Sam Smart sent a fierce drive just wide of Charlie Searle's far post after seven minutes while James Roberts flashed a dangerous effort across the face of goal at the other end less than a minute later as both sides offered an early threat.
Then Dan Bradshaw's deflected strike went the wrong side of the post for Eastleigh after striker Danny Whitehall's cross had picked him out following more good work to start the move from Smart on 18 minutes.
Hawks striker Scott Rendell then spurned a big chance for the visitors two minutes later, heading Oxford United loanee Leon Chambers-Parillon's cross over from close range.
But it was the Spitfires who took the lead 12 minutes before the break. Hawks were complaining about what they thought was a foul in the build-up, but Whitehall curled in brilliantly off the post with the outside of his right foot.
Searle then pulled off a fine stop with his feet after 41 minutes from Whitehall after he had been sent clear by Seung-Woo Yang as it stayed at 1-0.
Hawks came out with real impetus after the restart with Nicky Bailey seeing a low drive well saved by Flitney and Rendell's flicked header going just wide from Billy Clifford's corner.
Still the visitors continued to probe and came close to a leveller on 74 minutes. Substitute Tommy Wright rounded Flitney having latched onto a raking Paul Rooney pass but the angle closed on the striker with no-one able to get on the end of his ball into the box.
With Hawks pushing for a leveller, Eastleigh countered with Smart appearing to be tripped in the area, yet substitute Simpson picked up the loose ball and slotted home to wrap things up.
Hawks: Searle; Read, Rooney, Oastler, Newton, Chambers-Parillon, Bailey, Clifford, Widdrington, Roberts, Rendell. Subs: McCarthy (Chambers-Parillon, 49), Baggie (Widdrington, 56), Wright (Bailey, 63), Gobern, Worner.