Though born in Berkeley, California, Jennings was raised in West Des Moines, Iowa. There, he attended Valley High School. Playing for the school's baseball team, he led the team to back-to-back state championships in 2004 and 2005. As a senior in 2005, Jennings pitched to a 9–0 win–loss record and 0.85 earned run average (ERA), earning All-State honors.
Jennings attended the University of Nebraska, where he played college baseball for the Cornhuskers. He was drafted by the Marlins in the ninth round of the 2008 MLB draft.
He made his professional debut that season for the Jamestown Jammers of the Class A-Short Season New York–Penn League.
Across three levels of minor league baseball in 2009, Jennings had a combined 1–2 record, six saves, and 2.15 ERA in 45 games overall, allowing with 26 walks and while striking out 69 batters in 62⅔ innings pitched for the Greensboro Hornets of the Class A South Atlantic League, the Jupiter Hammerheads of the Class A-Advanced Florida State League, and the Jacksonville Suns of the Class AA Southern League. Baseball America rated Jennings as the Marlins' fourth best pitching prospect, and 13th best overall. Returning to Jacksonville in 2010, Jennings was suspended for 50 games for violating minor league baseball's policy banning the use of performance-enhancing drugs. The Marlins added Jennings to their 40-man roster after the 2011 season to protect him from the Rule 5 draft.
Jennings began the 2012 season with the New Orleans Zephyrs of the Class AAA Pacific Coast League. In nine appearances, he had a 2.08 ERA.
The Marlins promoted Jennings to MLB on April 30, 2012, and he pitched a scoreless inning in relief in his debut that day.
On August 7, 2014, Jennings was pitching in relief against the Pittsburgh Pirates when Jordy Mercer hit a line drive that accidentally struck Jennings in the head, inflicting a concussion in the process. Jennings was kept overnight in a Pittsburgh hospital for observations instead of accompanying the Marlins to Cincinnati for their next series against the Cincinnati Reds after the game. Jennings later thanked Pirates fans on Twitter for their support after the crowd at PNC Park gave Jennings a standing ovation when he was carted off the field and gave the crowd a thumbs-up.
The Marlins traded Jennings to the Chicago White Sox on December 11, 2014, in exchange for André Rienzo.
On July 27, 2017, the White Sox traded Jennings to the Tampa Bay Rays in exchange for first baseman Casey Gillaspie.
Jennings has been confused for the son of Dan Jennings, who was an executive for the Marlins when the younger Jennings was pitching for them. His parents, Bob and Janet Jennings, live in West Des Moines. Jennings has a brother, Tom, who currently works as a tennis professional at the Quad City Tennis Club. He is married. Jennings' parents, fiancee, and her parents all flew to Miami to watch Jennings' MLB debut.