John Paul Kissock opened the scoring with his debut goal for the club, while Keith Keane netted his first strike since the 8-0 hammering of Hayes & Yeading on March 27, 2010.
Brabin said: “We were all disappointed after Tuesday as we know we’re better than that, but I thought the lads showed a good attitude and played some really good stuff.
“I thought we deserved the win and it’s nice to get that.”
Town were excellent in the first half but only led by a single goal at the break, only for Keane to score direct from a free kick midway through the second period.
Brabin added: “I mentioned at half time, the fact we were only 1-0 up after dominating the first 25 minutes.
“Gateshead made more of a game of it, but I thought we created enough to go in a bit more comfortable than 1-0.
“For 15-20 minutes of the second half the game was tight, there wasn’t much in it. Then we started to get our noses in front and Keith scored at the right time.
“It was a really good performance. I thought we got stronger and stronger. We had another couple of chances but I’m delighted with performance, delighted with the result, and delighted to be in the semi-final now.”
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