The Nunholm men trooped off the New Williamfield ground to order their celebration pints at the bar – only to be told they hadn't actually finished the job.
Beveridge said: "It was all due to the scoreboard operators not co-ordinating with the official scorers. At the start of the final over the board indicated that we needed 13 to win – and thanks to three boundaries and a single, we reckoned we hit the target and everyone headed for the clubhouse.
"Then we saw the scorers scuttling across frantically to tell us that the totals were level and that we would have to go back out into the middle. It meant all the pressure was on tailender Jerry Goodin, who was facing – and thankfully he managed to hit the single we needed to win all over again."
While Goodin's final shot proved decisive, Beveridge picked out the man at the other end – skipper Stuart Corbett-Byers – for the main plaudits, after he finished on 72 not out.
"It was an amazing knock by Stuart, probably the best I have seen from him," he said.
Beveridge also insisted Dumfries have not yet got fully into their stride, adding: "We still have lots to work on. For a start, we have been giving away far too many sloppy extras and we need to address that quickly."
Former Dunfermline player Vasu Reddy had hit 74 to help propel Stirling County to 265 for five.
He then struck four times with the ball to give the home team the edge going into the tense finale, but Corbett-Byers and Goodin held their nerve to make sure of the last-gasp triumph.
It was not such a close encounter at Bothwell Castle Policies, where a 127 by Calum MacLeod inspired Uddingston to their demolition job on struggling Greenock. MacLeod faced just 97 balls, clearing the boundary twice and hitting the rope 14 times.
West of Scotland, meanwhile, proved they are no one-man team as they brushed aside the challenge of Drumpellier at Coatbridge.
World Cup player John Blain had been the dominant figure in their attack, but Iain Cran and Stewart Leggat combined for a six-wicket haul to dismiss Drumpellier for 155 in reply to 179 for six. West veteran Dougie Lockhart had laid the foundations with a trademark 59.
Clydesdale completed a comprehensive 127-run success against Ayr at Titwood with a solid, all-round team effort.