The hosts had gone in front four minutes from the break thanks to Conor Sammon, who netted from close range, drilling into the bottom left corner.
Kilmarnock had Alexei Eremenko sent off nine minutes into the second half, and he was followed down the tunnel within 11 minutes by visiting players Scott Robertson and Jon Daly, who each received two yellow cards in quick succession.
Sammon regained his starting place in the Kilmarnock line-up after turning down a move to Scunthorpe United.
Manager of the month Mixu Paatelainen made two changes to the home team.
Rugby Park's top scorer, who came off the bench in his side's 2-1 midweek defeat by Hearts, was included along with Jamie Hamill, who returned from suspension, while James Fowler and Mehdi Taouil dropped to the bench.
United goalkeeper Dusan Pernis recovered from a thigh injury to take his place in goal after missing Wednesday's penalty shoot-out victory at Ross County in the Scottish Cup.
It was one of four changes made by visiting manager Peter Houston - Craig Conway, Scott Robertson and Barry Douglas, all of whom came off the bench in the cup tie, started in place of the injured Paul Dixon and benched Garry Kenneth and David Robertson.
The lively Alexei Eremenko tested the United defence on just two minutes when he ran from the halfway line to the edge of the area and won a corner with his shot, the set piece being headed over by Frazer Wright.
Liam Kelly headed a cross narrowly wide of the left-hand post; while Sammon was then unable to direct on target a diving header from Garry Hay's promising cross.
Pernis was called upon to produce an impressive save on 25 minutes - Rui Miguel cut inside through the United defence, from right to left, and his eventual shot saw the goalkeeper stretch to tip it over the bar for a corner.
Goodwillie was about to celebrate as his low diagonal effort from the right looked certain to sneak in at the opposite post but, just as the ball appeared set to trundle across the line, it touched the upright and rolled on the goalline, allowing Hamill to pop up at the last possible opportunity and dramatically sweep it to safety.
Liam Kelly and then Sammon had further efforts before Kilmarnock took full advantage of their earlier reprieve by taking the lead five minutes before the break.
Sammon was in possession and bearing down on the left-hand side of the box, but elected to square across goal instead of shooting; the ball was then sent back across the box by Eremenko and Sammon lashed a close-range effort into the bottom left corner.
Kilmarnock were reduced to 10 men just 10 minutes after the restart when Eremenko was shown a straight red card for his reaction to a midfield foul by Scott Robertson, the Dundee United player also being booked the incident.
It was 10-man Kilmarnock who regrouped and had the next opportunity - Mohamadou Sissoko made an excellent run up the right and found Sammon in the centre; the lone striker held up the ball for the arrival of Miguel on the left, but he blazed his shot over.
As tempers remained high, United managed to see their numerical advantage become
a one-man deficit within 11 minutes of Eremenko's dismissal.
Just before the hour, Robertson received a swift second yellow card for a crunching challenge on Miguel close to his own penalty box; and six minutes later, he was followed to the away dressing room by captain Jon Daly, earning his second booking for the presence of an elbow as he challenged Sammon just 10 minutes after his first yellow card for dissent in the aftermath of the Eremenko incident.
Hamill hit the woodwork when he had a chance to double the hosts' lead from a good move as the game entered its final quarter.
Rui Miguel found Sammon on the right corner of the box and crossed for his team mate, only for Hamill's effort to rebound from the bottom of the left-hand upright.
Manuel Pascali then tried an ambitious lob forward to Sammon, with which he connected but was unable to direct his effort on target.
United's nine men worked hard in the latter stages, Goodwillie's heading over from a Barry Douglas corner seven minutes from time, and home goalkeeper Bell also holding a second set piece delivered by the same player after a penalty-box scramble.
The visitors were rewarded with an equaliser less than 30 seconds from full-time, substitute Kenneth netting with a speculative shot from distance.