Neither side was able to score during 120 minutes of tense action at Celtic Park but the Dons then tucked away all four of their spot-kicks, Adam Rooney putting the last one down the middle to clinch victory.
Inverness acquitted themselves with great credit in their first major final but they could not get the break through against an Aberdeen side who did not concede a goal in the entire competition until the shoot-out.
The match began on a balloon-strewn pitch and Aberdeen had to make a very early change when Jonny Hayes unfortunately injured his shoulder after taking a heavy fall when challenging Josh Meekings for a high ball and youngster Cammy Smith was called from the bench after just six minutes.
The Dons then came within inches of scoring in the 25th-minute when Caley keeper Dean Brill produced a weak punch and Russell Anderson crashed a shot against the inside of the post which the officials correctly ruled had not quite made it over the goal-line before Richie Foran managed to nudge it away.
Anderson headed down the subsequent corner but Adam Rooney put a reasonable chance over the crossbar.
Penalty claims rejected
And Aberdeen then appeared to have strong claims for a penalty rejected when Rooney went down under pressure from Meekings after latching onto a Shaleum Logan cross.
The Dons hit the woodwork again with half-time approaching when Mark Reynolds sliced an attempted cross but Brill had it covered and the ball bounced away off the angle between post and bar.
Inverness failed to pose many problems in the first-half, their best moment coming when Marley Watkins sent over a dangerous low cross which Anderson cleared from just in front of his own goal.
However, Caley grew in confidence as the match went on and began to attack more after the break.
Aberdeen goalkeeper Jamie Langfield nervously spilled a high David Raven cross and Anderson was grateful to clear, before Langfield made ground to his left to block a long-range Greg Tansey free-kick.
But it was then Brill's turn to have a lucky escape as he tentatively pushed a Ryan Jack shot into the air, but the goalkeeper just about managed to punch clear before Rooney could head home.
Suitably encouraged, Aberdeen finished the 90 minutes strongly and Robson produced a dangerous effort from the edge of the box which took a slight deflection and looked destined for the bottom corner, only for Brill to pull off a fine save.
Foran got on the end of a corner at the other end but headed over the bar and his appeals that he was held back by Andrew Considine fell on deaf ears.
Meekings saves the day
And Niall McGinn of Aberdeen had a fantastic chance to win it in the last five minutes after latching on to a misplaced Tansey pass and cutting inside but Meekings did superbly to slide across and deflect the shot over the crossbar.
Extra-time was therefore on the menu and McGinn was first to threaten but he pulled his shot wide of the target from 30 yards out.
Tansey put a free-kick just over the bar at the other end, and Doran's low shot from distance was then tipped around the post by Langfield, Meekings heading the subsequent corner over the top.
But the half-chances dried up in the second -period of extra-time and, after Nicky Low and Doran had blazed free-kicks over the crossbar at either end, a penalty shoot-out was required to resolve matters.
Inverness quickly put themselves in trouble in the shoot-out, Langfield saving a weak effort from Billy McKay before Tansey put a poor spot-kick over the bar.
Nick Ross and Doran calmly converted to get Caley on the board but Robson, Low and Scott Vernon had all found the bottom corner for Aberdeen, setting the stage for former Caley striker Rooney to secure the trophy.