Danny Elliott's finish and and Alex Penny own goal were enough for the Pilgrims to cancel out Ashley Hemmings' sublime finish as United beat Kidderminster Harriers 2-1 at Aggborough.
Boston - who lost Scot Duxbury to a faciual injury midway through the first half - put in a dogged team display and will now travel to AFC Fylde on Sunday.
"I thought the performance was excellent, especially the first half," Cox said.
"I thought the application and desire to win a game was superb and I thought we played some really good stuff against a really good side.
"They (Harriers) have had a fabulous season and they're a really top side, but I thought we edged it tonight.
"I thought the first half performance was dominant and we missed the two easiest chances of the half
"The finish for their goal was superb, great individualism.
"The one thing about the boys is they adapted, regrouped and scored the other goal.".
Victory sparked wild scenes at full time with the Pilgrims celebrating with the 400-plus travelling fans
"The players deserved it and the supporters deserved it," Cox added.
"When I came here I realised the players were over here and the supporters over there.
"I want the whole club to become one - whole.
"I'm losing words to describe our fanbase.
"On a Thursday night, this a hard place to get to, to take as many as we did - not just turning up but using their voices is superb. That helps us in or quest."
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