Two quick-fire goals from Scott Pollock put the Pilgrims in charge before Liam Nolan’s penalty set up a tense finish.
“We had to really dig in towards the end, they threw everything at us,” said Culverhouse.
“I thought we were in control until the penalty and we lost control of the game after that.
“It's understandable because we had not won away from home so it's sgodo to get the monkey off the back.
“The boys performed brainly in the second half when they were throwing it all us. We really stood up and put bodies on the line. I'm really pleased for them.”
Culverhouse also felt Pollock had got what he deserved after a fine display.
“He is a really good player,” he added. “He likes the role behind the front two where he can make good runs.
“He took both goals really well and it's a perfect reward for him after a good performance.”
The win was marred by a hamstring injury to Luke Shiels, which is likely to rule the defender out for some time.
“The first suggestions are that its a bad one which is a shame as he's been a catalyst for the club,” said Culverhouse.
“We will get it assessed and take it from there. We have got to look at things now as it leaves us really short.
“We will have to look at the loan market and see who we can bring in.”
Shiels picked up his injury in the 16th minute as United struggled to get going.
Jake Wright had a goal ruled out for offside, before Pollock fired over in stoppage time.
But he was on the mark when his free-kick, possibly with the aid of a deflection, beat the keeper on 57 minutes.
It was 2-0 three minutes later when he headed home from close range.
But Telford sent up a tense ending when Nolan slotted home from the spot after Ben Pollock brought down Jamie Allen.
Harrry Flowers headed against the crossbar late on, with Rhys Davies tipping over Piggott's follow-up as the Pilgrims held on.