The Taysiders, who had it all to do following their 2-2 draw at Tannadice in the first-leg last week, failed to recover from a disastrous start at the Arena Khimki in Russia when they went behind in the second minute to an Igor Semshov drive from the edge of the box.
Alexsandr Kokorin headed in the second after 22 minutes before midfielder Artur Yusupov ended any hopes the Scottish club had when he raced clear of the visiting defence to slip the ball past United's Polish keeper Radoslaw Cierzniak.
The second-half was more even in terms of possession but two late goals by substitute Aleksandr Sapeta made it 7-2 on aggregate for the Russians to complete a miserable night for the Scots.
United had travelled to Moscow with some hope not only because of their encouraging display in the first-leg.
Dinamo remained at the bottom of the Russian Premier League after losing 4-0 to city rivals Spartak on Sunday, a result which ended Sergei Silkin's tenure as boss with Dmitri Khokhlov taking over caretaker manager.
However, the former Russia international saw his new charges get off to the perfect start when Marko Lomic swung a low cross in from the left for Semshov to rifle past Cierzniak.
The Tannadice men were exposed again when Hungarian winger Balazs Dzsudzsak got away from Keith Watson down the left and his cross to the back post saw Kokorin beat United defender Barry Douglas to head past Cierzniak.
In the 29th minute United's defence parted all too easily from a long punt by Dinamo keeper Roman Berezovsky to allow Kokorin to race in on goal but this time Cierzniak blocked the shot with his leg before making a better save at his top corner from Kevin Kuranyi moments later with the Taysiders' defence split open again.
Peter Houston's men were all over the place and were let off the hook when Kokorin headed over the bar from 10 yards when he ought to have at least hit the target.
However, the third goal was almost inevitable and it came in the 39th minute when Yusupov was sent racing through on goal by the quick-thinking Kokorin and he almost casually knocked the ball past the defenceless Cierzniak.
United looked shellshocked but they rallied in the closing stages of the first-half.
Two minutes from the break winger Gary Mackay-Steven had half a chance in the box but failed to control Willo Flood's cross from the right before a Flood free-kick from 25 yards forced Berezovsky into a good save.
United, however, soon had their backs to the wall again when the game restarted with Kuranyi having a couple of efforts on goal, a header wide and a shot which was saved by Cierzniak.
On the hour mark Kuranyi was denied the goal he was so clearly chasing by United defender Brian McLean's saving tackle.
The Tannadice side rallied a little after the home side had made a triple substitution and Richie Ryan had a long-distance effort which cleared the bar.
Out-of-sorts Kuranyi missed the target again from the edge of the box but Sapeta made it 4-0 from close range with eight minutes remaining before hammering in an unstoppable shot from 30 yards in the final minute of normal time.