Everton's supporters had grown restless after seeing their side given the runaround by the improving League One outfit, and were largely out-sung by nearly 6,000 visiting fans who had already started to dream of a money-spinning replay at Brunton Park.
But Tim Cahill and Leighton Baines sent the Merseyside followers home content, if not delighted, after netting in the final eight minutes for their disjointed team after goals created by rookies Seamus Coleman and Keiran Agard.
It had started well enough for the hosts, James Vaughan rounding Adam Collin for a 12th-minute opener after Adam Clayton's horrendous back-pass was easily intercepted, but the villain turned hero after playing a key part in their deserved equaliser six minutes later.
Tim Howard couldn't stop the Manchester City loanee's shot as he looked to round off a move he'd started in midfield, and Kevan Hurst added the decisive touch to poke the ball over the line.
Danny Livesey rattled United's best chance against the crossbar with 20 minutes to play as they almost stole victory through their tenacity, but the Toffees' Premier League quality was to prove vital.
With Everton's subs boasting just 24 club appearances between them, David Moyes had little to choose from with the game needing a lift, but he chose expertly with his young selections.
Coleman's fleet-footed play down the right-hand side reinvigorated the home side, and, when his cross was headed down by Diniyar Bilyaletdinov, Cahill was perfectly placed to sweep home at the second attempt after his first shot had been charged down by Ian Harte.
Baines then rolled home a stoppage-time penalty after Agard broke into the box at pace and was brought down by Collin, who was left with little option.