The 23-year-old, who will be starting in place of the injured Wade Davis, has already been with the Rays twice this season and has made five starts, going 2-0 with a 3.41 ERA.
His second stint in the majors was far smoother than his first, as he won two of his four starts and posted a 2.55 ERA. Still, he was sent back down to Triple-A on June 18 in order to make room for Jeff Niemann's return to the rotation.
The Rays may need a big start from Cobb tonight, after the team used nine pitchers in a 1-0 loss to the Boston Red Sox in 16 innings on Sunday.
The AL East rivals totaled 29 runs in splitting the first two games of the series before combining for a whopping eight hits in Sunday's five-hour, 44- minute finale -- the longest game in Rays history.
"I don't know what to say," Rays center fielder B.J. Upton said. "I think both teams pitched well. They had some opportunities, we had some opportunities, we were just one big hit away.
"I'd probably say this is the toughest loss of the year. To know that we played as well as we did and our pitching kept us in it the whole game. We couldn't scratch a run. ... I think a lot of us in the clubhouse right now are pretty much not happy."
Dustin Pedroia provided three of the Red Sox's five hits, none bigger than a single off Adam Russell (1-2) that plated Josh Reddick in the 16th.
Against a Boston club that amassed 47 runs over its previous six games, Niemann yielded a mere two hits and two walks while recording a career-high 10 strikeouts.
Tampa has dropped four of five and has fallen seven games back of the Red Sox in the division,
The Yankees, meanwhile, enter this series on a high note after winning the final two games of their four-game series with the Toronto Blue Jays at Rogers Centre. On Sunday, Curtis Granderson had two hits, knocked in three runs and scored once to help lead the Yanks to a 7-2 win.
Brett Gardner went 3-for-4 and crossed the plate three times as New York won for the fourth time in six games.
Phil Hughes (1-2) went six innings, allowing two runs on four hits and two walks while striking out five for the win.
"He was great today. He had good command and good composure out there," Yankees catcher Russell Martin said of Hughes. "He just looked confident."
Getting the call for the Yankees tonight will be right-hander A.J. Burnett, who is 8-7 with a 4.15 ERA. Burnett pitched well in his final start before the break against Tampa, but did not get a decision, as he allowed three runs and three hits in 5 2/3 innings of his team's 5-4 win. He is 0-4 with a 7.83 ERA in his last five starts against them.
Burnett has faced the Rays 27 times and is 12-8 with a 3.41 ERA in that span.
The Yankees have won three of five against Tampa Bay this season, including two of three at home before the All-Star break.