Undrafted, Ebbett played collegiate hockey with the University of Michigan of the CCHA. After captaining the Wolverines in his senior year, Ebbett was signed as a free agent by the Ottawa Senators on July 17, 2006. Andrew then made his professional debut in the 2006–07 season with Ottawa's affiliate, the Binghamton Senators of the AHL.
On May 16, 2007, Ebbett was signed by the Anaheim Ducks. He was then assigned to affiliate the Portland Pirates of the AHL but made his NHL debut in the 2007–08 season with the Ducks, playing in 3 games. On June 11, 2008, Ebbett was re-signed by the Ducks to a two-year contract extension.
In the 2008–09 season, Ebbett began the year with affiliate, the Iowa Chops, after 28 games with the Chops he was recalled and scored his first NHL goal against the Philadelphia Flyers on January 2, 2009, his 26th birthday. Ebbett established a scoring role with the Ducks scoring 32 points in only 48 games helping the Ducks to the second round and scoring his first Stanley Cup playoff goal in the upset of the San Jose Sharks on April 19, 2009.
Ebbett played 2 games with the Ducks in 2009–10 before he was claimed off waivers by the Chicago Blackhawks on October 17, 2009. Unable to find a role within the Blackhawks roster Ebbett was again waived after 10 games and acquired by the Minnesota Wild on November 21, 2009. He settled quickly scoring a goal in his first game with the Wild against the Boston Bruins on November 25, 2009. Ebbett remained with Minnesota and completed the season posting 8 goals for 14 points in 49 games.
On July 2, 2010, Ebbett joined his fourth team in less than a year when he signed as a free agent to a one-year deal with the Phoenix Coyotes.
On July 1, 2011, he signed a one-year contract worth $525,000 with the Vancouver Canucks.
On July 2, 2012, Ebbett signed a one-year, two-way contract with the Vancouver Canucks that paid him $600,000 at the NHL level and $300,000 at the AHL level.
On July 5, 2013, Ebbett signed a two-year, two-way contract with the Pittsburgh Penguins that paid him $1.1 million at the NHL level.
As an impending free agent following the 2014–15 season with the Penguins, Ebbett signed his first contract abroad in agreeing to a two-year contract with Swiss club, SC Bern of the National League A (NLA) on June 24, 2015. He helped Bern win the 2016 Swiss championship, scoring five goals and ten assists in 14 playoff contests. In the regular season, he had seen the ice 19 times with four goals and nine assists. In October 2015, Ebbett had fractured his tibial plateau and was sidelined for three months. In December 2016, he won the Spengler Cup with Team Canada and was named to the tournament's all-star team. In January 2017, he signed a contract extension with Bern that would keep him with the team until 2019.