The Turkish striker struck with the last kick of the ball to consign to defeat Danny Lennon's men, who were leading after first-half goals from Paul McGowan and Steven Thompson.
But Jackie McNamara's United side struck three times in the final 11 minutes with goals from substitute pair Brian Graham and Stuart Armstrong, setting Ciftci up to round things off in style.
The hosts were desperate to bounce back from their midweek defeat by St Johnstone.
But it was the Buddies who were first to threaten in the opening stages when full-back Sean Kelly burst down the left into some space.
Kelly sent over a tempting cross for Josh Magennis but he couldn't keep his header down and his effort went well wide of the target.
Shortly afterwards, McGowan's in-swinging cross picked out Thompson and his glancing header fizzed just inches past home goalkeeper Radoslaw Cierzniak's left-hand post.
However, it didn't take the visitors much longer to get their noses in front as they opened the scoring in simple fashion in the 19th minute.
Marc McAusland floated a free-kick from inside his own half into the United penalty box which did not look to be too troubling for the home defence.
But John Souttar stood and watched the ball come into the danger area and before he knew it, McGowan connected with a backward header which found its was past Cierzniak from 10 yards.
And they quickly set about trying to increased their lead by sensing the opponents were there for the taking.
McGowan was in the thick of things again after 23 minutes as he latched onto a loose ball which the United defence had failed to clear.
He only had one thing on his mind as he took a touch before unleashing a shot from the edge of the penalty area which Cierzniak did well to hold.
United were barley being seen in an attacking force although Gary Mackay-Steven was trying his best to change that.
But Saints struck again in the 39th minute when John McGinn's low cross from the left caused panic in the home back-four.
Gavin Gunning attempted to hook it away but succeeded only in hitting the ball straight off Thompson before the ball nestled in Cierzniak's corner from point-blank range.
Lacklustre United did have a chance to throw themselves a lifeline on the stroke of half-time.
Ciftci's delicate cross into the box fell perfectly for El Alagui, who was all alone inside the box, but the Frenchman hacked at his effort and could only watch as his left-foot volley blazed well over Marian Kello's bar.
At the start of the second-half, Mackay-Steven released Ciftci but the Turkish striker took one touch too many as he attempted to round Kello and the chance went begging.
Then Mackay-Steven minutes later struck a blistering 25-yard drive which Kello spectacularly turned over.
But the Arabs pulled one back in 79 minutes when Ciftci's pass releasing Ryan Dow, whose cross was turned home from close range by fellow substitute Graham.
And five minutes later they found themselves back on level terms through Armstrong.
Mackay-Steven's cross meant McAusland was not able to properly clear, so Armstrong was on hand to apply the finishing touch with a calm finish from 15 yards.
Then, in injury-time, Ciftci sealed things by bundling the ball home from close-range after Kello had spilled Dow's powerful shot to complete a magnificent comeback.