Verdasco, ranked 36th in the world, went a break up but was broken back twice by the American as he lost the first set 6-4.
The second went to a tie break after Kudla saved break and set point, with Verdasco losing his first three service points as the world number 122 prevailed 7/4.
Zverev beats Verdasco to reach quarter-finals in Cologne
Frances Tiafoe made it through to the next round, beating wildcard Mackenzie McDonald 6-3 6-3 in one hour and seven minutes.
Tiafoe broke his opponent twice for 5-3 and served to love for the first set, and broke the world number 142 early in the second.
He pressed home his advantage, winning the match on his first break point at 5-3.
Earlier in the day, Tennys Sandgren made short work of Blaz Kavcic taking just over an hour to see off the Slovenian 6-2 6-2, Henri Laaksonen took less than 60 minutes to book his place in the next round with a 6-4 6-3 win over Dustin Brown and Bjorn Fratangelo beat Jordan Thompson 6-1 6-1.
At the Grand Prix Hassan II in Marrakech, Gilles Simon disposed of number six seed and fellow Frenchman Benoit Paire 6-2 6-7 (7/1) 7-6 (7/5).
Paire, ranked 47th in the world, was broken on his first service game and again for 4-2 with Simon taking the set on the second break and set point.
In the second, both players broke each other three times before Paire broke four times in the tie-break to force a decider.
Simon saved match point at 5-4 in the third, and won the next three points as the final set went into a tie-break where Simon won six points in a row.
Richard Gasquet is also in the next round, beating Thomas Fabbiano 6-1 7-6 (7/2) in one hour and 29 minutes.
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