United midfielder Ryan Dow hit the post before Saints defender Steven Anderson, in his testimonial year, headed in a David Wotherspoon corner as a pulsating first half moved into stoppage time.
Tannadice striker Nadir Ciftci struck the bar with a free-kick early in an occasionally tetchy second half and the Saints had to defend for their lives at times before striker Steven MacLean ensured history would be made with a second after 83 minutes to send their 15,000 fans - a third of the population of Perth - into raptures.
The McDiarmid Park side were appearing in their first final of the competition and few outside Tannadice will begrudge them the trophy, albeit they rode their luck at times against a side they had beaten three times in four Scottish Premiership meetings this season.
The biggest Arabs support ever of close to 30,000 fans who made the journey from Tayside, and who helped create such a colourful spectacle in the east end of Glasgow, slipped out after the final thinking of what might have been in their second final in four years.
Amid the frenetic, full-blooded opening minutes in which United looked the more likely, Saints goalkeeper Alan Mannus nervously fumbled a cross by Dow but no-one in orange was in position to take advantage.
Neither side could or would put their foot on the ball, with possession changing hands at times by the pass.
In the 10th minute 27-goal Saints' striker Stevie May had a speculative shot from distance which failed to bother United goalkeeper Radoslaw Cierzniak but which showed the highly-rated 21-year-old was in confident mood.
Cierzniak was made to work moments later when Wotherspoon knocked the ball back to fellow midfielder James Dunne just outside the box. The playmaker's strike was true but was tipped over the bar for a corner, from which Saints defender Frazer Wright failed to make a decent connection from Brian Easton's delivery..
There was an unsuccessful claim for a United penalty when Dunne lashed the ball against what look like Wright's arm and there was another let-off for the Perth side in the 32nd minute when Dow's clever flick from an Andy Robertson cross beat Mannus but rebounded off the post before being cleared for a corner which came to nothing.
United ended the first half on top but found themselves behind by the time they ran up the tunnel.
The PA announcer at Celtic Park had just revealed that there would be a minimum of one added minute when Anderson rose at the back post above Keith Watson at the corner to power his header past Cierzniak, setting off an explosion of blue in the stands.
Saints, though, survived another close call at the start of the second half after Dunne was booked for a foul 20 yards out.
Ciftci's curling free-kick beat the defensive wall and Mannus but came down off the bar and on to the back of the Irishman, who gratefully grabbed it inches from the line.
There was even more drama at the other end moments later when May had the ball in the net at the second attempt with his hand, for which he was booked, with the Perth forward claiming his header had crossed the line.
It was breathless, end-to-end action.
Wotherspoon's long run into the United box ended with his shot parried by Cierzniak and, when United broke, Mannus had to push Robertson's drive round the post for a corner, which the McDiarmid Park side survived.
Mannus then had to race from his goal to foil Dow and then Watson in the same move.
Lee Croft replaced Michael O'Halloran for Saints before Brian Graham came on for Paul Paton, the United striker firing wide of the target from 14 yards within minutes of his 77th-minute introduction.
As the tension increased another notch, Ciftci worked a yard of space for himself 25 yards from goal before drilling his right-footed shot just wide of the post.
However, with seven minutes remaining, Saints got the clincher.
MacLean found himself the beneficiary of a deflected May pass to go clean through against Cierzniak and, although the United keeper got a block on the first effort, the striker knocked in the loose ball while lying on the ground, before running to take the acclaim of the fans, who knew the wait to see the Scottish Cup coming to Perth was over.
14th minute - Dundee United keeper Radoslaw Cierzniak does well to tip over fierce volley from James Dunne
32 - The Terrors' Ryan Dow hits the post with a clever flick and watches in frustration as it runs along the goal line before being booted to safety
44 - Big United shout for a penalty as St Johnstone's David Wotherspoon appears to handle in the box. Referee Craig Thompson, however, waves play on.
45 - The Perth side grab the lead with last kick of the first half. Cierzniak comes to take Wotherspoon's deep corner but gets nowhere near it as Steven Anderson powers home a superb back-post header.
50 - Nadir Ciftci so close to an equaliser for United with a free-kick which hits the bar but lands in the grounded Alan Mannus' grateful grasp.
53 - Saints striker Stevie May bizarrely punches the ball into the net when he could have kicked it in. Rightly booked by Thompson.
82 - Ciftci is close again as he rifles a low drive an inch wide of Mannus' right-hand post.
84 - Steve MacLean wins it for Saints. A lucky ricochet puts the frontman one-on-one with Cierzniak, but while the keeper saved the first effort he could do nothing as MacLean bundled home the rebound.