Lancers were made to pay for three glaring first-half misses and, without the injured Shaheen Sadough, who failed a late fitness test, they struggled for height and a physical presence up front.
It looked as though they would hang on for a replay though, until Ampthill striker Lea Coulter scrambled home Ed Wheatley’s free kick three minutes from the end of the additional 30 minutes.
Then, with Lancers throwing everyone forward in desperation, Ampthill sealed their passage to the national section through substitute Kyle Duncan.
Lancers joint manager Jason Rutherford admitted his team did not create enough and that he feared for them as soon as Sadough was ruled out.
He said: “You can’t miss the chances we did and win, it’s a missed opportunity and very disappointing, but that’s football.
“We didn’t create enough to hurt them, they were decent but I think we would have been favourites in the replay.”
Lancers started brightly and saw a golden chance fall their way on 12 minutes. Wayne Joseph got free down the left and crossed for Darren Annis, who dragged wide from six yards when it looked easier to score.
Tyrone Taylor lashed over for Ampthill but Lancers went ahead in fine fashion on 23 minutes when Jack Hayward curled home from 20 yards after Darren Boswell’s corner.
They should have gone further ahead 10 minutes later when Annis again was guilty of squandering a glorious opportunity.
Good link-up between Hayward and Graham Martin saw the striker played clean through, but, with just the keeper to beat, he hesitated and fired wastefully wide from seven yards.
Lancers were made to rue that moment five minutes later when Danny Nicholls glanced home a header from Taylor’s corner to bring Ampthill level.
Annis then did put the ball in the net only to be flagged for offside, and Graham Martin should have restored Lancers’ lead but headed over from close range from a pinpoint Boswell cross.
After the interval, Nicholls and Coulter both had efforts well stopped by Lancing keeper Tony Di Bernardo, while Lancers’ Adam Hunt headed against the bar and Sam O’Regan curled a good effort narrowly wide.
Boswell went close again and Dan McLoughlin flashed an effort over in a tightly-contested second half, but neither side could find the breakthrough.
In extra-time, Duncan fired a warning shot when he struck the post on 104 minutes after Nicholls played him in, but with a replay imminent first Coulter, then Duncan, hit the two late strikes to leave Lancers wondering what might have been.
LANCING: Di Bernardo; Taylor, O’Regan, Gainsford, Hunt; Hayward, McLoughlin; Boswell, Martin, Joseph; Annis. Subs: Paterson (Annis 63), Turner (Hayward 98), Lucas (Hunt 117), Sharman, Evans. Att: 136.