Rankin netted just six minutes after Curtis Good had levelled for United, with Alexei Eremenko and Kris Boyd scoring for the visitors after Gary Mackay-Steven's early opener.
United took the lead after 10 minutes when Mackay-Steven pounced on a mistake by Jackson Irvine and kept his composure to slide the ball past the advancing Craig Samson.
Eremenko pulled Kilmarnock level five minutes before the break when Rory McKenzie headed a poor Radoslaw Cierzniak goalkick into his path and the diminutive playmaker dispatched a superb shot from 30 yards.
Boyd (64) put the visitors ahead when he beat Cierzniak to the ball as the keeper advanced from his penalty area and his lob bounced past Andrew Robertson, who looked certain to clear off the line.
A minute later Mackay-Steven's corner found Good and the United defender directed his header into the top corner.
Rankin settled the match on 71 minutes when he blasted into the roof of the net from close-range from Ciftci's pass, seconds after Samson has denied the midfielder at full stretch.
The home side made an impressive start as striker Ciftci (5) had an effort come back off the post, and nine minutes later Mackay-Steven cut inside his marker and fired just wide of the target.
That chance came moments after Lee Ashcroft arrived at the back post to head into the corner of the United net, but his he was denied a goal by the assistant's flag.
Kilmarnock continued to grow in confidence though and Cierzniak produced a brilliant save to deny Boyd during a goalmouth scramble before McKenzie hit the post with the rebound.