Just when time looked to be running out, Kacie Elson equalised in the 86th minute to force the additional 30 minutes against their visitors, who play in the league above United.
However, despite the hard work of Becky Langley's squad, Brighouse scored two goals in extra time to ultimately emerge victorious.
Reacting to the loss, Elson said: "It feels gutting to know we were just short of moving onto the next round. We gave everything and nobody can fault any of our efforts.
"We were unfortunate with our chances, and we needed that little bit more composure to be clinical and punish their mistakes as they did with ours."
Despite being disappointed with the result, Elson maintained that there were many positives for the Lady Magpies to take.
"We dominated throughout the 120 minutes," she insisted. "We didn't stop working and kept our identity on and off the pitch. I believe we played some great football and worked for each other, creating some great chances.
"To compete with a team who play in the league above and make them work for the result should be a compliment to what we as a team have been working towards all season."
She added that Langley will not be allowing her squad to dwell on the loss and championed the way her manager prepares the team for each fixture.
"Becky will put this result in the past," Elson added. "She won't let the feeling of disappointment linger in this week's training, and we will work on being prepared for the FA Cup this weekend."
Mental strength is something that is synonymous with Langley's squad, and Elson praised the way the United women fought back against Brighouse to get their late leveller.
She said: "We had been working so hard for the equaliser and to score it was a relief. It gave us momentum at such an important moment, and I think we all had the mentality and belief that we were the better team and would go on and win the game."
The Lady Magpies' next fixture is in the Vitality Women's FA Cup, when they host Harrogate Town Ladies on Sunday, 14th November. Looking ahead to that fixture, Elson outlined her target.
She said: "Our aims are to work endlessly for each other. We want to ensure we get the win and ensure that we don't concede."
Photograph courtesy of Colin Lock/@Tyneside_NUWFC
