On their sun-drenched home pitch, United looked set to be undone by another wonder goal after Kilmarnock's summer signing Rory McKeown broke the deadlock a minute after half-time.
Peter Houston's team were knocked out of Europe on Thursday night after Sebastian Dudek of Slask Wroclaw curled home superbly from 25 yards - and left-back McKeown's effort was equally eye-catching from an even greater distance.
But Swanson ensured a share of the spoils with a sweet strike of his own 20 minutes from time.
United, who lost their Europa League second round qualifying tie on away goals to the Polish side, set out seeking to improve on last season's fourth-place league finish.
Garry Kenneth and Scott Severin dropped out from Houston's midweek team, with Barry Douglas and 19-year-old midfielder Stuart Armstrong brought in.
A much-changed Kilmarnock, now under the permanent charge of last term's caretaker boss Shiels and new assistant Jimmy Nicholl, started with five of their eight summer signings.
McKeown, Patrick Ada, Gary Harkins, Zdenek Kroca and Ben Hutchinson were all handed competitive debuts.
United made the more purposeful start, dominating possession as Kilmarnock struggled to find any kind of early rhythm.
Former Hibernian midfielder John Rankin registered the first effort for the hosts after eight minutes, trying his luck from long range on the half-volley but his strike flew just wide.
Rankin was soon involved again, cutting the ball back from the left, but David Goodwillie could not connect and team-mate Armstrong was unable to put sufficient power on a shot to overly trouble Cammy Bell.
It was 17 minutes before a serious threat came from the Ayrshire outfit and they almost profited.
Captain Manuel Pascali fed James Dayton in a central position and the former Crystal Palace youth-team player surged toward goal, leaving Paul Dixon in his wake, only to drag his angled shot inches wide.
Goodwillie should have put United ahead after 20 minutes. Dixon's inviting cross from the left landed at the feet of the in-demand striker six yards from goal, yet somehow he put the ball the wrong side of the post.
The Taysiders were showing no signs of a European hangover, with skipper Jon Daly next heading off-target from a Keith Watson flick-on.
As the game became stretched, Harkins brought a first save from Dusan Pernis as the Slovakian clutched his 22-yard effort.
Kilmarnock suddenly had their tails up and former Dundee midfielder Harkins had two close calls in quick succession.
First his close-range shot was blocked by Pernis from a Dayton corner, then - as the loose ball bobbled around - his follow-up effort was blocked on the line by his own player, Pascali.
Johnny Russell and Goodwillie combined at the other end a minute before half-time but Kroca was well placed to divert the latter's effort around the post.
The visitors stunned United by making the breakthrough in grand style after the re-start.
McKeown, the Northern Ireland Under-19 international released by Ipswich this summer, collected a loose ball from a Kilmarnock corner around 30 yards from goal and fiercely drove his shot past Pernis.
It was a wonderful strike, with the ball appearing to move in the air to outfox the United 'keeper.
Daly nodded a Sean Dillon cross wide as Houston's charges sought an immediate reply, before Ada brilliantly challenged Goodwillie to deflect his chance wide for a corner.
Houston had made all three of his substitutions just after the hour mark as he sought to pile the pressure on the away defence.
Cameroon defender Ada next blocked a net-bound Daly header as the onslaught continued.
Dayton fired an angled shot wide as Kilmarnock made a rare foray forward, but the equaliser was only delayed until the 70th minute.
Swanson, who had only been on the field for 13 minutes after replacing Russell, initially struck a tame free-kick into the visitors' wall.
But the 24-year-old made no mistake with the rebound, driving low past Bell from 20 yards, before earning a booking for taking his shirt off during a gleeful celebration.
Ada was again well-placed to deflect yet another Goodwillie strike away from goal, before the dangerous Dayton once more tried his luck from an angle.
Two minutes into injury-time, Goodwillie - coveted by clubs in Scotland and England - was handed the perfect chance to further enhance his reputation.
Collecting a pass from Daly, the 22-year-old was clean through on goal but as he tried to go wide of Bell the 'keeper clawed the ball from his feet.