Drewiske attended High School in Hudson, Wisconsin. His accomplishments included 1 state title and was the recipient of the Wisconsin High School Mr. Hockey award in 2003 having successfully recovered from a neck fracture in the previous season.
Undrafted, Drewiske started his junior career with Des Moines Buccaneers of the USHL before joining the University of Wisconsin in the Western Collegiate Hockey Association. Drewiske captained and tied for first on the team in goals for defenseman in his last season with the Badgers.
On April 1, 2008, Drewiske signed an entry-level contract with the Los Angeles Kings. He was then assigned to affiliate, the Manchester Monarchs of the American Hockey League for the end of the 2007–08 season. Drewiske started the 2008–09 season with the Monarchs before receiving his first call up to the Kings on February 2, 2009. Drewiske made his NHL debut in a 1-0 win against the Ottawa Senators on February 3, 2009. His first NHL goal was an empty net goal to seal a 6-4 win on October 6, 2009 against the San Jose Sharks. He won a Stanley Cup in the 2011-2012 season.