Verdasco exploited Murray's fragile serve to win 6-4, 6-4 in a match which finished after midnight German time. Verdasco broke Murray's serve four times -- once to love and three times to 15.
Murray lost in the second round of the U.S. Open and in the first round of the French Open. He used wild cards for both of those Grand Slam tournaments and again in Cologne for his first loss on an indoor hard court since the 2015 tour finals.
Verdasco ended a four-match losing run going back to January in the Australian Open. It was the Spaniard's second match since the tour resumed amid the coronavirus pandemic, his first being a one-sided loss to Damir Dzumhur in qualifying for the Rome Masters last month.
Verdasco will take on top seed Alexander Zverev in the second round after the German had a bye in the first round.
Fellow Brit Kyle Edmund was also eliminated after losing a tight match 7-5, 7-6 (1) to big-serving South African Lloyd Harris, who hit eight aces. Harris' reward is a second round match with U.S. player Steve Johnson.
No. 8 seed Croatian Marin Cilic had to dig deep to avoid a first round elimination as he beat Marcos Giron 6-2, 4-6, 6-3. Giron had lost in qualifying and was a late replacement after Cilic's initial opponent Ricardas Berankis withdrew.