Barry Douglas, Conway and Swanson netted as United climbed into fifth place in the Clydesdale Bank Premier League, with Josh Magennis replying for the Dons. Conway and Swanson were substituted in the second half as Houston had one eye on a packed programme - Monday night's match was the first of five for United in 12 days.
"I've always said if I can get Conway and Swanson on the park at the same time we'll cause teams problems and we did cause them problems," said Houston.
Conway's swerving strike, which perhaps should have been stopped by Dons goalkeeper Jamie Langfield, handed the hosts a 2-0 half-time advantage before Magennis reduced the deficit after 51 minutes.
But Swanson struck a superb third after 60 minutes as Houston's men moved a point ahead of Inverness and above Motherwell on goal difference with three games in hand on their rivals.
Having played fewer matches than anyone else in the SPL with the split approaching, United are playing catch up with fixtures against Hamilton on Thursday, Well in the Scottish Cup on Sunday and league games with Hearts and Inverness next week.
Houston is poised to rotate his squad, with players returning to fitness to boost his resources - including Mihael Kovacevic, who was in the squad for the first time since the Scottish Cup win at Hampden last May.
"We want the points and we need the points to make sure we consolidate in a top-six finish," added Houston. "I've not got a massive amount of players to change it that much, but certainly we'll have to use in the next couple of days the full squad."
Aberdeen had gone from relegation contenders to top-half hopefuls after being revitalised since Brown's appointment in December, but the former Scotland boss was frustrated by his side's display.
Brown said: "I think that's the worst disappointment I can remember since we came to Aberdeen."
Source: PA
Source: PA