Scott Vernon put the Dons in front with a tidy finish from Ryan Fraser's cutback early on, but United levelled through Gavin Gunning's brave effort.
An own-goal from Aberdeen goalkeeper Jamie Langfield saw the Tangerines go in front before Aberdeen's top scorer Niall McGinn thundered home a free-kick to ensure the points were shared.
Aberdeen are beginning to see the light at the end of the tunnel after working their way through the last six weeks missing the bulk of their midfield, and Brown hopes his side can kick on as players return.
"I'm very optimistic for the second half of the season. We're getting some players back, which will increase the competition for places," Brown said.
"I personally don't need to achieve anything, but I think Aberdeen would like to achieve. What I would like to do, every year, is improve on the previous year.
"At the moment we're improved on last year. We're still in the Scottish Cup, having beaten the second best team in Scotland in the previous round.
"We're still in touch in the league; nobody's going to catch Celtic, but we're in touch with the battle for second, third and fourth.
"We're aiming for the highest possible place in the SPL and I think we're capable, if we get everybody fit and a break or two.
"What we would like to achieve is a good finish in the SPL and a good run in the cup. That would be what the fans here deserve. We'll do all we can to please the Aberdeen support."