After a disappointing first half, Pavol Safranko opened the scoring with a close-range header.
Scott Harrison prevented the Bairns from falling further behind when he cleared off the line from Yannick Loemba before Deimantas Petravicius missed Falkirk's best chance 60 seconds later.
Victory was confirmed by United when Craig Curran nodded a looping header over Leo Fasan.
The results leaves the Bairns rock bottom of the table with no points from six games and in deep trouble, Dundee United move to third in the table.
Falkirk came into the match on the back of a dispiriting 3-2 defeat to Ayr United last week after leading the Honest Men 2-0 at the break.
They started on the back foot in this match and Csaba Laszlo's men almost opened the scoring when Christoph Rabitsch thundered a volley wide of the target.
That set the tone for the first half with United in control and dominating for large periods. At the other end, chances were few and far between.
Zak Rudden, who scored last week, went close with a header on 14 minutes for the Bairns, but that was about as good as it got except for a Tom Dallison first-half header.
At the other end, the pressure continued to mount with Curran's volley being deflected wide.
Despite United enjoying the bulk of possession, they weren't able to turn that into meaningful chances. They did come close before the break when Paul McMullan was released down the right wing and his cross broke to Loemba who fired harmlessly wide.
Falkirk needed to improve on their first-half showing and Petravicius showed his intent with some neat play down the left before firing in a powerful, low drive which stung the palms of Rakovan.
They suffered a blow when United took the lead through Safranko. Aird delivered a long, hopeful ball from out wide and the Slovakian striker beat Fasan in the air, to nod it into the empty net.
United almost doubled their lead as Loemba did brilliantly to engineer space in the box before lashing a shot at goal though Harrison was there to clear off the line. Falkirk almost levelled on the counter but Petravicius couldn't turn the ball home from close range.
The visitors doubled their lead and extinguished any hope of a Falkirk revival when Curran sent a loping header over Fasan from Loemba's delivery.