The home side were on top before defender Robbie Neilson, previously booked for a foul on Samaras, was shown a straight red in the 28th minute for leading with an elbow in an aerial tussle with the Greece striker.
The game turned on that incident and in a one-sided second half the home side were put to the sword.
Joe Ledley, Georgios Samaras and Anthony Stokes racked up an unbeatable lead before Hoops skipper Scott Brown scored from the spot a minute from time after Samaras had been tripped by Sean Dillon in the United box.
Celtic remain on course for the treble but United will wonder what might have happened had the experienced Neilson not left referee Iain Brines without an option.
Manager Neil Lennon made four changes to his in-form side for the trip to Tayside.
Brown returned from a groin injury with right-back Adam Matthews, midfielder Victor Wanyama and Samaras back in the starting line-up.
Kelvin Wilson and Kris Commons dropped to the bench with Mikael Lustig and Andre Blackman out altogether.
Celtic players wore black armbands as a mark of respect for the late QC Paul McBride, who died last Sunday and who had previously represented the club.
United, unchanged and on the back of three successive wins, started confidently and in the eighth minute skipper Jon Daly got past Hoops stopper Charlie Mulgrew to get on to the end of a Willo Flood cross but headed past the near post from eight yards.
The United striker then had a weak claim for a penalty when he went to ground in the box after a challenge by Parkhead stopper Thomas Rogne but Brines stood firm.
Meanwhile, Lennon had moved the powerful Wanyama from midfield to centre-back, with Mulgrew going to left-back and Joe Ledley stepping up to midfield.
The Clydesdale Bank Premier League champions-in-waiting were working their way slowly into the game but United looked the more dangerous.
In the 25th minute wide-man Gary Mackay-Steven went past Matthews down the left before crossing for Russell, whose first time shot from 12 yards was parried by Celtic keeper Fraser Forster with Daly failing to hit the target with the rebound.
Three minutes later the game took a dramatic twist when Neilson had his moment of madness.
There was little room for complaint from the United camp and Samaras resumed after treatment.
Houston brought on defender Garry Kenneth for Mackay-Steven but immediately Celtic were on the ascendancy and it was the home side who were forced into some desperate defending to keep their goal intact before the break.
The Taysiders started the second half on the back foot with Celtic sensing the first goal could open the floodgates.
Daly ploughed the lone furrow for United but was mostly reduced to a watching brief, although in the 51st he did manage a snap-shot from 20 yards but it left Forster untroubled.
But almost inevitably Celtic forged ahead with the goal coming through Ledley.
The Wales international was involved at the start of the move, and when striker Anthony Stokes went past United defender Dillon and cut the ball back, Ledley pounced from six yards to whip the ball past Tannadice keeper Dusan Pernis at the second attempt.
United were rattled and moments later Kenneth bravely stuck his head in the path of a Samaras drive from the edge of the box which appeared goalbound.
In the 63rd minute, with the visitors in complete control, Stokes tried his luck with a low drive from 20 yards but Pernis gathered with ease.
Five minutes later, in a rare break, United substitute Milos Lacny, on for Gavin Gunning, burst into the Celtic box and had only Forster to beat but failed to connect with his shot and the ball trundled past the post.
Three minutes after that, though, Samaras secured his side's place in the semi-finals when he rose inside the United box to head in a looping Mulgrew free-kick.
With the Hoops fans still celebrating, Russell flashed a drive from 14 yards high over the bar when he should have hit the target.
In the 79th minute Stokes missed a sitter when he rattled the bar from eight yards after being set up perfectly by James Forrest, but there was time for the Irishman to make amends, latching on to a Ledley pass with four minutes remaining before slipping the ball past Pernis.
A minute from time Samaras won a spot-kick when he was tripped by Dillon inside the box and Brown converted the penalty to rub salt into the wounds of United, who were left bedraggled.