Murray had won their one previous match emphatically on a hard court in San Diego last year, and he had early control of the semi-final this time at the Challenger Tour event.
However, Murray could not force a second-set breakthrough and lost the tie-break on the way to a 5-7 7-6 (7-4) 6-3 defeat.
The Scot, a title winner at Wimbledon in 2013 and 2016, served 15 aces to Kudla's 12. Murray won just 45 per cent of points on his second serve, though, an area where he will hope to improve as the grass season progresses.
He heads to Stuttgart next week before playing at Queen's Club in the run-up to Wimbledon.
World number 81 Kudla, who reached the third round at Wimbledon last year before losing to Novak Djokovic, said: "This is definitely, probably, the biggest win of my career.
"The conditions were super windy, but in the warm-up I felt I was hitting the ball clean."
Kudla will face Australian Jordan Thompson in Sunday's final and added: "There's something about England and grass, I just feel comfortable and at ease. I feel like a different tennis player after being terrible on clay for two months."
Kubler and Verdasco endured a 71-minute first-set tussle, which the Australian edged 9-7 in a tie-break before dominating the second frame, dishing out a bagel in 29 minutes to win 7-6 (9-7) 6-0.
World number 71 Kubler progressed to the second round where he will take on Colombia's Daniel Elahi Galan who defeated China's Zhang Zhizhen 6-2 6-3.
Johnson set up a second-round clash with American compatriot and top seed Frances Tiafoe after getting the better of Facundo Bagnis 7-6 (7-5) 1-6 6-1 in one hour and 55 minutes.
USA's Kudla also progressed with a hard-fought 6-4 4-6 6-4 victory over Australia's Aleksandar Vukic in just under two hours.
Dutchman Gijs Brouwer won 6-4 6-4 over Aleksandar Kovacevic, setting up a meeting with big-serving fourth seed John Isner.
Coming off an epic three-hour duel Wednesday in his first match since the French Open final, Nadal fell to Lloyd Harris 6-4 1-6 6-4 on Thursday at the Citi Open.
After a short-lived maiden appearance at the Washington, D.C. tournament, the Spaniard will look ahead to Toronto and Cincinnati as he continues his preparations for the U.S. Open.
Harris, meanwhile, heads to a quarter-final matchup against veteran Kei Nishikori after recording the biggest win of his career.
Facing Nadal for the first time, the 24-year-old South African slammed 16 aces Thursday and saved four of six break points.
The 20-time grand slam winner, who battled a foot injury to outlast Jack Sock in his first match, did not have enough to answer the world number 50 on Thursday.
In addition to the top-seeded Nadal's defeat, second seed Felix Auger-Aliassime also went down Thursday, falling 6-3 6-4 to American wild card Jenson Brooksby.
With their exits, the tournament is now without its top four seeds.
That leaves fifth seed Jannik Sinner as the top player remaining after he defeated Sebastian Korda 7-6 (7-3) 7-6 (7-3).
Nishikori advanced by downing seventh seed Cameron Norrie 3-6 6-3 6-3, while John Millman knocked out eighth seed Reilly Opelka 6-3 7-6 (7-4).
Among unseeded players, Denis Kudla defeated the red-hot Brandon Nakashima 3-6 7-6 (7-3) 6-4, while Steve Johnston eased past Ricardas Berankis 6-2 6-1 and Mackenzie McDonald beat Illya Ivashka 6-4 6-4.
In addition to Harris-Nishikori, Kudla will face McDonald in the other quarter-final in the top half of the draw, while Sinner meets Johnson and Millman takes on Brooksby in the bottom half.
By far the most experienced player remaining in the field after Harris' upset of Rafael Nadal on Thursday, Nishikori prevailed 6-3 7-5 to avenge a loss earlier this year in their only prior meeting.
Nishikori had lost his last six ATP quarter-final matches dating to a trip to the semis in Barcelona in April 2019.
The 31-year-old next faces Mackenzie McDonald, who downed countryman Denis Kudla 6-3 6-2 to reach his second career ATP semi-final. He has never played in a final.
In the other semi-final, fifth seed Jannik Sinner will meet wild card Jenson Brooksby.
Sinner had little trouble with American veteran Steve Johnson in a 6-4 6-2 win as the Italian won 93 per cent of his first-serve points and saved all three break points he faced.
Brooksby maintained the strong return game that as carried him all week in a 6-1 6-2 upset of 11th seed John Millman.
The Australian did not manage an ace against the 20-year-old American, who won 54 per cent of the points on Millman's serve and converted five of seven break points.
At 130 in the world, Brooksby is the lowest-seeded Citi Open semi-finalist since John Isner reached the 2007 final while ranked 416th.
Nishioka was the highest seed in action on the first day of the tournament, and he had his back up against the wall after failing to win a game in a troubling second set, but produced three breaks of serve in the back-and-forth decider.
Meanwhile, Ecuador's Emilio Gomez earned a shot at top seed and world number seven Taylor Fritz after advancing 7-6 (7-3) 6-4 against Taiwan's Tung-lin Wu.
Portugal's Nuno Borges continued his winning run after earning his spot through the qualifiers, eliminating America's Steve Johnson 6-4 3-6 6-4, but the crowd got something to cheer for when the USA's Denis Kudla got the better of Australia's Jordan Thompson 3-6 6-4 7-6 (7-2).
Further south at Argentina's Buenos Aires Open, Serbia's Laslo Djere booked a blockbuster showdown against world number two Carlos Alcaraz after repelling the challenge of Italy's Fabio Fognini 6-4 6-4.
Argentinian Pedro Cachin put on a show for his hometown fans with a 7-6 (7-2) 6-4 victory over Brazil's Thiago Monteiro, but Cachin's compatriot Guido Pella had less success with a 6-4 7-5 defeat at the hand of Spain's Jaume Munar.
The man with the Golden Slam in his sights took out Kudla in straight sets on Court One, but a 6-4 6-3 7-6 (9-7) victory was not entirely routine.
Qualifier Kudla broke the Serbian's serve twice and kept it a largely tight contest, showing the form that took him to the fourth round six years ago.
After roaring to glory at the Australian Open and French Open, Djokovic could become just the second player in history to win all four grand slams and an Olympic gold medal in the same year, after Steffi Graf's 1988 feat.
There can be no easy sauntering to glory on that scale, so after a pair of relatively easy wins over Jack Draper – albeit after a first-set hiccup – and Kevin Anderson, this was more of a taxing clash that could benefit Djokovic in the long run.
Kudla was gifted a 3-0 lead in the third-set tie-break as Djokovic twice double-faulted, but he could not capitalise, chopping a pair of poor drop shots on the way to allowing his opponent to gain the upper hand.
A solid overhead gave Djokovic a first match point, which the five-time Wimbledon champion squandered with a forehand into the net. He soon had a second though, and Djokovic, defending his 2019 title this fortnight after last year's tournament was cancelled, displayed stunning defence on the baseline before Kudla netted on the forehand.
The world number one then spoke of how growing up during the Yugoslav Wars in the 1990s had hardened him for a career in sport, when asked at courtside what gave him such a drive to win.
"I think part of it is genes. My family, the way I've grown up in difficult times during the nineties for my country," Djokovic said on the BBC.
"Failure was never on option, For me or anybody for my family. We had to find a way to find the basic needs for us to survive. During those times it was difficult and that has strengthened my character, I would say.
"Part of it also comes from my upbringing in the mountains. I spent a lot of time in the mountains with wolves, so this is a wolf energy."
Data Slam: Shaky serves as Djokovic does enough
Kudla is no stranger to SW19 success, given his 2015 run that was ended by Marin Cilic. And with Djokovic only landing 54 per cent of first serves in court across the contest, that might have opened a door for the American here. Yet Kudla was also struggling with his first delivery, putting just 47 per cent of his first serves into play, and few could hope to beat Djokovic with such a low percentage.
WINNERS/UNFORCED ERRORS
Djokovic – 34/28
Kudla – 30/34
ACES/DOUBLE FAULTS
Djokovic – 8/6
Kudla – 6/3
BREAK POINTS WON
Djokovic – 4/10
Kudla – 2/4
