Born in Grindavík, Alfreð played for Ungmennafélag Grindavíkur's youth teams while also spending two years of his childhood in the Scottish capital Edinburgh.

[4] Alfreð played for boys' club Hutchison Vale, and became a fan of Hibernian, while his father studied in the city.

[14] On 6 March 2012, Alfreð joined Swedish champions Helsingborgs IF on a six-month loan deal, ending on 15 August 2012.

[23][24][25] He made his debut for the club later that month, starting in a 2–0 home win against Aberdeen for the season's UEFA Europa League.

[26] Greek champions Olympiacos had previously tried to sign the 26-year-old Icelandic, but failed to reach an agreement with his former club, Heerenveen.

On 1 September 2022, Finnbogason signed a one-year contract with recently promoted Danish Superliga club Lyngby.

[35] Following Eupen's relegation to the Challenger Pro League, Alfreð reached an agreement with the club to terminate his contract on 5 September 2024.

His first goal came on 8 September 2009 in a 2011 UEFA European Under-21 Football Championship Group 5 qualifier 6–2 win against Northern Ireland under-21s at The Showgrounds, Coleraine.

[38] Alfreð won his first cap for the Iceland national team in 2010, coming on as a substitute in a friendly match against the Faroe Islands.

[39][40] Alfreð scored on his full debut for Iceland against Israel in a 3–2 loss at Bloomfield Stadium in Tel Aviv.

The previous record-holder had also played in the Netherlands, Pétur Pétursson, who scored 23 goals in 33 matches for Feyenoord in the 1979–80 Eredivisie.