Sævarsson started his professional career with local team Valur in the Úrvalsdeild. He went on to play 66 competitive leagues games for the club, scoring twice.
In 2008, at age 23, he joined the Norwegian side SK Brann in Tippeligaen. Being known as a versatile defender, he was originally planned as a right back at Brann, but played almost all the 2009 season on the left back. He went on to play a total of 6 seasons for the club before signing for Hammarby IF in December 2014, when Brann got relegated to the second tier.
At Hammarby he began the 2015 season as a centre-back, but eventually moved back to his natural position on the right flank. After putting on impressive performances on both positions, he was voted as the "newcomer of the year" by the supporters.
Birkir was selected for Euro 2016 in France. In the second group stage game against Hungary he scored an own goal resulting in a 1-1 draw.