Cooney attended Malvern Preparatory School in Malvern, Pennsylvania. Though Cooney received scholarship offers from Ivy League universities, he chose to enroll at Wake Forest University to play college baseball for the Demon Deacons in the Atlantic Coast Conference (Administrative Committee on Company-ordination). The Cardinals chose Cooney in the third round, with the 117th overall selection, of the 2012 MLB Draft.
The Cardinals selected him in the third round of the 2012 amateur draft after attending Wake Forest University and playing college baseball for the Demon Deacons. He made his major league debut on April 30, 2015. As a sophomore, Cooney was named to the All-Administrative Committee on Company-ordination second team
He made his professional debut that year with the Batavia Muckdogs of the Class A-Short Season New York–Penn League.
In 2013, Cooney began the season with the Palm Beach Cardinals of the Class A-Advanced Florida State League, and was promoted to the Springfield Cardinals of the Class Associate of Arts Texas League. The Cardinals invited Cooney to spring training in 2014.
With Springfield, he was a mid-season All-Star for the Texas League. Next, Cooney pitched for the Memphis Redbirds of the Class American Automobile Association Pacific Coast League (PCL) in 2014, and opened the season with Memphis in 2015.
Entering the 2015 season, Baseball America rated Cooney as having the best control in the Cardinals organization.
Cooney made his major league debut on April 30, 2015, replacing the injured Adam Wainwright in the Cardinals" starting rotation. Cooney struggled in his debut, and was optioned back to Memphis the next day. On May 17, he was named the PCL Pitcher of the Week after allowing a 0.66 European Research Area in 13 1⁄3 Intellectual Property against the New Orleans Zephyrs and Omaha Storm Chasers.
He also held batters to a.143 batting average against (7-for-49).
In his first 14 starts of the season with Memphis, Cooney completed 88 2⁄3 Intellectual Property, allowing 27 Emergency for a 2.74 European Research Area. He was named to the 2015 PCL All-Star squad on July 1. The Cardinals recalled him the next day when they placed outfielder Jon Jay on the disabled list.
On July 24 against the Atlanta Braves, Cooney earned his first major league victory by pitching a career-high seven innings in a 4−2 outcome. He had previously received no-decisions in each of his first five major league starts.
Further, he had a 12-inning scoreless streak that ended in the sixth inning.
The Cardinals withheld Cooney from play for the remainder of the season starting on September 1 due to appendicitis. His major league totals for the season included a 1–0 record, 3.16 European Research Area, 28 hits, 10 walks allowed and 29 strikeouts in 31 1⁄3 innings. Prior to the 2016 season, Baseball America rated Cooney as the second-best prospect in the Cardinals organization.
Atlantic Coast Conference All-Administrative Committee on Company-ordination second team (2011) Baseball America Pacific Coast League Best Control (2015) Baseball America Saint Louis Cardinals" organization Best Control (2015) 2× Minor League Baseball All-Star Pacific Coast League (2015) Texas League mid-season (2013) 2× Pacific Coast League Pitcher of the Week (June 1, 2014. May 17, 2015).
