Russell struck after just 16 seconds before Jon Daly doubled the advantage and Russell put the gloss on the victory, which saw Kal Naismith and Ian Black dismissed late on.
Controversy surrounded the build-up to the tie when Rangers opted against taking a ticket allocation for the match.
The decision was a response to calls from fans groups to boycott what was the first away fixture against Scottish Premier League opposition since the club were denied entry to the top-flight after being consigned to liquidation in the summer.
Of the 9,564 crowd inside Tannadice, 365 were recorded as visiting fans, with some Gers supporters opting to take in the match regardless - although they may have been wishing they had not bothered making the trip to Tayside in the end.
McNamara enjoyed the best possible start to his tenure when the home side surged into the lead with just 16 seconds on the clock.
A defensive error by Emilson Cribari allowed Russell to nip through on goal and dispatch a shot beyond the out-stretched hand of Neil Alexander.
United had the net bulging again after 36 minutes when Daly took advantage of poor defending to rise unmarked in the box and nod home from Willo Flood's free-kick.
There was no way back for Rangers when Russell claimed his second - and United's third - of the day with 10 minutes to go. Michael Gardyne nodded down for Daly and he played into the path of Russell to sweep into the back of the net from six yards.
A miserable day was compounded for Rangers when Naismith was shown a straight red for a challenge on Flood after 86 minutes, before Black picked up a second caution for a foul on John Rankin a minute later.
Paul Heffernan's double sent Kilmarnock into the quarter-finals with a 2-0 win over Inverness, who suffered their second cup exit in eight days.
The substitute fired home in the 58th minute after Inverness were again exposed on the break, although they suffered an unfortunate run of the ball.
And the Irishman's clever back-heel with five minutes left ensured Caley Thistle have now lost as many games in those eight days as they had in the rest of the season, with their Scottish Communities League Cup semi-final defeat by Hearts followed by a league defeat at St Mirren.
Esmael Goncalves continued his wonderful start at St Mirren with a brace in their 2-0 win against St Johnstone.
The 21-year-old took his goal tally to four in three games as the Buddies booked their place in the last eight.
Goncalves headed home to break the deadlock six minutes after half-time and doubled the hosts' advantage eight minutes later.
Hamilton ensured their progress into the sixth round of with a 2-0 win over First Division rivals Dunfermline at East End Park.
Steven May opened the scoring when he fired into the net from just outside the box five minutes after the restart, before Jonathon Routledge's right-footed shot from inside the box found the top corner on the hour mark.
Falkirk eased into the quarter-finals with a convincing 4-1 win over Irn-Bru Second Division side Forfar.
Lyle Taylor took his tally for the season to 22 in all competitions with a brace. He put the First Division outfit ahead in the 19th minute with a close-range finish following good work from Blair Alston.
But Forfar were level two minutes later. Stephen Kingsley impeded Ross Campbell in the area and Iain Campbell powered his penalty into the bottom corner.
Falkirk re-took the lead in the 68th minute as Stewart Murdoch headed in Sean Higgins' corner.
Seven minutes later substitute David Weatherston gave his side a two-goal cushion with a low shot into the bottom corner from 12 yards. And Taylor added his second seven minutes from time with a close-range volley from Higgins' cross.