Niall McGinn scored the winner with a fine 18th-minute solo effort to leave the visitors languishing at the foot of the table.
Brown claimed his side could have had a first-half penalty which might have made the game more comfortable, and said: "The second half from our point of view was poor; the first half was good.
"I thought we played well, and there was a clear handling offence on Scott Vernon's shot which would hopefully have given us a more comfortable lead.
"The performance wasn't good in the second half, and it kind of makes up for a fortnight ago against Hibs. We dominated that game and only got one point
whereas in the second half tonight Dundee dominated and we got three points, so it perhaps evens it out a bit."