United slipped out of the competition after a 3-2 defeat on penalties in a match which they had led.
Jaanai Gordon had fired the Pilgrims into a 20th minute lead, only for Alex Bradley to equalise 18 minutes after the break.
Sam McLintock and Tom Platt scored from the spot, but once Jas Singh had denied Will Atkinson and Jake Wright, Sam Long crashed United’s fifth spot-kick over the bar to seal Tamworth’s triumph.
“We wanted to go further in the competition and we just have to get on with it,” he said.
“We were good in the first half and restricted them to hardly anything.
“In the second half we got embroiled in the game they wanted to play where it was just headed back down our throat and we couldn’t get the ball down.
“I thought we were comfortable in the first half, but we eased off in the second and let them come on to us.
“We have got to be a bit cuter in the way we are. We are young but we have to be more streetwise in how we deal with game management.
"We had some good passages of play in the first half and moved the ball really well. There are positives we can take out of it.”
Culverhouse also had some words of comfort for Long after missing the decisive penalty.
“On the day it is who feels confident and who wants to take them,” he said. “Then it is down to the goalkeepers.
“Sam is fine, it's another learning experience for a young player and that is why they are here.
“He wanted to come through and take one. He was confident and that is good on him for a 19 year-old kid.”
Elsewhere, Boston United midfielder Joe Leesley has returned from his loan spell at Darlington – to complete a permanent transfer to Kidderminster Harriers.
Leesley scored once in seven appearances for the title-chasing Quakers during his month in the northeast – and has now put pen-to-paper on a contract at Aggborough after the two clubs reached an agreement.
The 28-year-old made 66 appearances in total for the Pilgrims – scoring nine goals in the process.