Marcus Trescothick's 80 from 54 balls was ultimately in vain as Somerset were bowled out for 267 to lose to Surrey by seven runs by the Duckworth/Lewis method following a floodlight failure at the Kia Oval.
Chasing Surrey's 50-over total of 284 for nine, Somerset set out believing they needed to overhaul that score in no more than 42.1 overs in order to win a quarter-final place.
Trescothick's assault, which included four sixes and nine fours, left them on 146 for four after 24.1 overs, and very much on course to get to their target within a target, when a power surge in south London blew out the Oval floodlights.
A 25-minute delay, while a fuse was repaired and the floodlight bulbs were allowed to cool down before they could be turned on again, hurt Somerset as when the game restarted their adjusted Duckworth-Lewis target to 275 in 46 overs meaning they needed to get there in 38.2 overs.
James Hildreth and Alex Barrow returned to the field knowing just 14.1 overs were available to score the 129 more runs required for victory, and guaranteed last eight qualification.
Hildreth did his best by reaching 53 from 46 balls but at 184 he drove to extra cover and when Barrow and Craig Overton also fell in the next two overs Somerset were suddenly 195 for seven.
Tim Groenewald edged Tim Linley behind at 222 in the first of the powerplay overs and when Luke Gregory's 19-ball 23 ended in the 36th over, Somerset's quest for qualification was seemingly over.
Nottinghamshire looked as if they might do Somerset a favour, though, by failing to beat Warwickshire, meaning that all Somerset needed to do was win this game.
However, after a brave stand of 34 by last pair Alfonso Thomas and Jack Leach, the news of Nottinghamshire's dramatic last-ball victory came just before the latter was run out for 18 by a direct hit by Aneesh Kapil at cover.
It brought to an end a remarkable match which had begun with Surrey making only a moderate start after Trescothick had won the toss and, unsurprisingly, chose to bowl first.
Jason Roy was leg-before for 10 to Gregory in the second over, to a ball slanted into his pads, and Vikram Solanki lost his off stump to Peter Trego's third ball in the 10th over as Surrey reached the end of the initial powerplay at 56 for two.
Steven Davies went to a 48-ball half-century with six fours before missing a reverse-sweep off left-arm spinner Leach and departing lbw in the 19th over.
Rory Burns also batted well until he was caught off a leading edge for 40 and it was left to Gary Wilson, who had given a difficult chance to a diving Johann Myburgh at midwicket when on eight, to give the innings renewed momentum in a 74-run partnership in 13 overs with Kapil.
Wilson hit Leach for six over long-on and also hit five fours in his 45 from 59 balls before being run out by Overton in the last over of the batting powerplay.
Kapil continued to flourish and his run-a-ball 59, which ended when he sliced Groenewald to cover, was followed by quickfire contributions from both Zafar Ansari and Robin Peterson.
Somerset's reply was launched in thrilling style by Trescothick, who drove Linley over long-off for six in a ninth over costing 17 and then pulled George Edwards for successive sixes in a 10th over that brought 19 runs.
There was a fourth six for Trescothick, swatted off Peterson, but by then Somerset had lost Nick Compton for 22 and Trego for five, leg-before to a Jade Dernbach full toss.
Ansari won lbw appeals to send back both Trescothick and Myburgh just before the floodlight failure.