Osweiler chose to attend Arizona State University over scholarship offers from Stanford and Washington State. As a true freshman in 2009, Osweiler played in six games with one start. He became the first true freshman to start a game for the Sun Devils since Jake Plummer in 1993. He finished the season completing 24 of 55 passes for 249 yards with two touchdowns and two interceptions. As a sophomore in 2010 he again played in six games with one start. For the season he completed 62 of 109 passes for 797 yards, five touchdowns and no interceptions. As a junior in 2011, he took over as the Sun Devils starting quarterback after the retirement of Steven Threet. He finished the season with 4,036 yards and 26 touchdowns.
Osweiler measured 6'6 and 7/8 inches tall at the 2012 NFL Combine, instead of the 6'8 that media had touted throughout his Arizona State career.
Osweiler was selected with the 57th overall pick in the second round of the 2012 NFL Draft by the Denver Broncos. He signed a rookie contract for four years and $3,516,000 through the 2015 season. He made his NFL debut in Week 4 against the Oakland Raiders. The Broncos won the game 37–6. He threw his first pass in a week 17 victory over the Kansas City Chiefs, finishing 2 of 4 for 12 yards. During the Week 12 overtime loss in the 2013 season, Osweiler was put into the line in an attempt to block the New England Patriots game-winning field-goal, as he is the tallest member of the team at 6'7" (201 cm). Osweiler threw his first career touchdown against the Oakland Raiders on December 28, 2014.
Against Kansas City on November 15, 2015, Osweiler took over for Peyton Manning, who was benched after throwing four interceptions and posting a passer rating of 0.0. The Broncos later announced that Osweiler would start in place of an injured Manning the following week against the Chicago Bears. In his first start on his 25th birthday, he completed 20 out of his 27 passes for 250 yards with two touchdowns and a 127.1 passer rating in a 17–15 win over the Bears, becoming the first player to ever get and win his first career start on his birthday. Osweiler was presented the game ball after the game by head coach Gary Kubiak. The next day, it was announced that Osweiler would start the following week against the New England Patriots. On November 29, 2015, Osweiler led the Broncos to a win over the then-undefeated Patriots in overtime, 30–24. He completed 23 of 42 passes for 270 yards, one touchdown, and one interception.