The 2010 winners went ahead after 16 minutes when John Rankin fired in his first goal for the club and despite enduring a few nervy moments the visitors put the tie to bed just before half-time when Scott Robertson side-footed home the second.
Russell extended the visitors' advantage 17 minutes into the second half with a fine finish high beyond Connor Fairley and the Scotland Under-21 international got his second of the game six minutes later as he headed home Paul Dixon's cross.
Gary Mackay-Steven made it five with another excellent strike before Russell completed his hat-trick six minutes from time. However there was some solace for Airdrie as Ryan Donnelly scored twice in the final five minutes to bring some respectability to the scoreline.
The hosts had defeated Gala Fairydean 11-0 in the previous round and their bright start suggested they were in the mood to avenge a 2-0 defeat to the same opponents in the Scottish Communities League Cup earlier this season.
Jamie Stevenson saw his deflected effort from the edge of the area fly just over nine minutes in and from the resultant corner the Airdrie midfielder came close again, this time seeing his shot come back off the bar.
However the Second Division side were soon undone on when Mackay-Steven found Rankin bursting into the box and he fired powerfully past Fairley.
Willie McLaren passed up a great chance to equalise for the home side when he volleyed straight at Dusan Pernis midway through the half and the Slovak was again on hand to deny Stevenson as Airdrie pressed forward.
It was the visitors though who were demonstrating the greater quality in the final third and in particular the combination of Dixon and Mackay-Steven down the left. Dixon's low cross just evaded the on-rushing Russell and Jon Daly four minutes before the break but Airdrie didn't heed the warning and the second goal arrived just a minute later.
Mackay-Steven's excellent cut-back found Russell who laid the ball into Robertson's path and the midfielder expertly guided the ball into the top corner.
The second half followed a similar pattern as after a promising start to the half Airdrie conceded again. This time Russell found acres of room inside the area to control Willo Flood's cross before crashing an excellent finish into the roof of the net.
The gulf in quality was now starting to tell and the Clydesdale Bank Premier League side added another two quickfire goals as Russell headed home another excellent Dixon cross before Mackay-Steven got the goal his performance deserved with an accurate shot across Fairley.
Russell then completed his hat-trick as he slotted home the rebound after Fairley had blocked his first effort and there was even time for Airdrie to score twice as Donnelly first rounded Pernis and slotted into the empty net before heading in at the far post with the last touch of the game.