Aberdeen lost to First Division opposition in both domestic cups last term, beaten by Dundee in this competition and Raith Rovers in the Scottish Cup, but Alloa had no such surprise in them and contributed to their own downfall.
Robertson's dismissal for bringing down Fraser Fyvie also resulted in a penalty put away by Paul Hartley to open the scoring and Rory McArdle claimed a second for Aberdeen before half-time with an easy header.
Any lingering hopes of an Alloa fightback disappeared on the hour mark when Chris Maguire's long-range strike made it three.
Alloa started with positive intent and might have taken an early lead when Scott Walker's firm header from John Grant's cross went narrowly wide of the post.
The home team were to make life difficult for themselves, though, from the 15th minute onwards, a chain of events put in motion by a dreadful attempted backpass from Billy Gibson.
The Alloa defender almost seemed to slip on the ball and it rolled invitingly for Fyvie to burst clear, and after he rounded Robertson the goalkeeper brought him down for the double punishment of a red card and a penalty kick.
Alloa midfielder Kevin McDonald was the innocent victim as he had to make way for substitute goalkeeper Thomas Flynn.
Hartley, restored to the Scotland squad earlier today, stepped up to fire in his fourth penalty of the season, having claimed a hat-trick of them on the season's opening day against Hamilton.
Aberdeen should have doubled their lead after 26 minutes when Fyvie darted down the left this time and squared for Maguire who missed an open goal with a dreadful miss.
Hartley went for an audacious chip that had Flynn backtracking as it looped just over the bar, but Aberdeen were to make it two before the break.
Hartley's corner was met by Josh Magennis and while Flynn saved, no Alloa defender could complete the clearance and McArdle nodded into an unguarded net.
A minute before the break came a potentially serious injury to 18-year-old Fyvie, who seemed to twist his knee in the artificial turf near the touchline with his manager Mark McGhee looking accusingly at the surface.
Hartley was replaced by Scott Vernon after 49 minutes, but Aberdeen retained control without him and extended their lead to 3-0 with a speculative shot from distance from Maguire that Flynn might have done better with.
Maguire almost scored again with a neat turn and shot after good set-up play from Andrew Considine.
There was another injury scare for Aberdeen when Zander Diamond limped off, reducing the visitors to 10 men as they had already used their full allocation of three substitutes.
Stuart Noble had a late consolation chance for Alloa, but his effort was straight against the legs of Mark Howard in the Aberdeen goal.