The Dark Blues had captain Gary Irvine sent off just 20 minutes into the derby encounter at Tannadice but took the lead through Ryan Conroy's 67th-minute goal.
However they were denied a famous victory in the final minute when Michael Gardyne equalised for the hosts.
Brown said: "To play for 70 minutes with 10 men, we had to defend very well.
"We knew we were not going to get many opportunities but we scored a fantastic goal.
"We caught them on the counter-attack and we hoped we would see it out but Dundee United are a good side and we were at their ground. I just felt for our players at the end.
"At the sending-off, Gary led with his elbow but he just went to win the header.
"I cannot fault the referee but I was a defender myself and when you are using your arms for leverage, there is a fine line between dangerous play.
"I thought the referee had a good game so I cannot complain about it.
"We have pegged a point and we wish it was three. After the sending-off you would probably say you would settle for a point but we got the goal and we are just disappointed we could not see it out.
"We were unlucky but it is good for the players in the dressing-room at the end that they were deflated despite being down to 10 men."
Dundee are yet to lose in the SPL since Brown took charge last month and they now trail St Mirren by 13 points at the foot of the standings.
Their prospects of staying up remain bleak, yet Brown said: "We will keep going until someone tells us we are down. We have plenty of spirit and showed that again here today.
"We took a point against a good Dundee United side for the first time this season so that can only be positive."