Cole's record ended at 61 strikeouts before issuing a walk to Joey Gallo in the third inning.
His remarkable stretch began on April 12 against the Toronto Blue Jays, when Cole walked Lourdes Gurriel Jr. at the start of the second inning. It was just the third walk that Cole had issued on the season at the time.
He now has five walks on the season, after walking two batters against the Rangers.
Milwaukee Brewers righty Corbin Burnes set the previous MLB record at 58 strikeouts without a walk earlier in the 2021 season, with his streak coming to an end on May 13 against the St. Louis Cardinals.