The hosts looked to have earned all three points after James Dayton's stunning strike early in the second half, however Dixon hit back in injury-time to ensure Peter Houston's men left with a share of the spoils.
United started the match full of confidence after battling back to defeat Inverness last weekend and they created the first chance of the match, John Rankin forcing Cammy Bell into a good low stop.
A minute later United came even closer to opening the scoring when Jon Daly connected with Gary Mackay-Steven's corner but his header was turned on to the bar by Bell.
Buoyed by their impressive start the visitors again carved out an excellent chance after six minutes, but again Bell produced an excellent save to deny Daly.
It took the hosts 12 minutes to create their first chance of the encounter. The Killie fans shouted for a penalty when Dixon appeared to fell Dayton inside the area but the ball broke to Dean Shiels, who unleashed a stinging effort which struck the crossbar.
Minutes later Manuel Pascali passed up a glorious chance when he blazed wide from eight yards after clever build-up play from Rory McKeown.
The game continued to swing from end to end and the woodwork again came to Kilmarnock's rescue when Stuart Armstrong's 20-yard drive beat Bell but came back off the inside of the left-hand post.
The best chance of the first 45 minutes fell to United captain Daly. Johnny Russell evaded two Kilmarnock defenders and threaded the ball through to the onrushing striker, who slid the ball wide of the post with the goal at his mercy.
Bell was playing a pivotal part in keeping his side on level terms and after the home defence failed to deal with Mackay-Steven's corner, the Scotland international made a smart save at the feet of Armstrong.
After a dull opening to the second half the game was brought to life as Kilmarnock grabbed a sensational opener.
United lost possession in the middle of the park, allowing Dayton to drive forward and unleash an incredible 25-yard strike beyond keeper Dusan Pernis and into the top left-hand corner.
Two minutes later Russell passed up a chance for a quickfire leveller when the ball broke to him on the edge of the area but he dragged his shot wide of the target.
Former Blackburn defender Gavin Gunning also failed to hit the target when he connected with Dixon's free-kick on 56 minutes as the visitors struggled to turn their possession into goals.
Unlike the first half, both teams found goalscoring opportunities at a premium.
Goal hero Dayton, after a tireless performance, was replaced by David Silva as the hosts looked to hold on to their lead.
On 80 minutes United forced Bell into his first save of the second half, when Mackay-Steven unleashed a tricky shot from just inside the penalty box.
The Kilmarnock defence were rarely threatened by the visitors' attack but as the clock reached 90 minutes they were dealt a crushing blow.
Armstrong fed Dixon, who cut inside and fizzed an unstoppable effort low beyond Bell.