Born in Limanowa, Zbozień began his career as a youth player for Zyndram Łącko. Then he played for two years in Sandecja Nowy Sącz youth teams. In 2007 Zbozień joined Legia Warszawa and on 30 July 2007 he made his Młoda Ekstraklasa debut in a 3–1 loss to Cracovia.
Three months later, on 4 December 2007 he made his debiut for Legia first team in a 2–0 victory over ŁKansas Łódź in Ekstraklasa Cup.
On 7 April 2008 Zbozień scored his first goal in a 2–1 victory over GKS Bełchatów in Młoda Ekstraklasa. He ended the 2007-2008 season with 27 appearances in that competition and one in Ekstraklasa Cup.
On 27 August 2008 Zbozień played for Legia reserves in a 0–3 loss to Motor Lublin and he ended the 2008-2009 with 28 appearances and four goals in Młoda Ekstraklasa. He also appeared in one match in Ekstraklasa Cup.
In March 2010, he moved to Sandecja Nowy Sącz.
In June 2011, he joined Legia Warszawa. In February 2014 he joined the Russian side Football Club Amkar Permanent, where he met the fellow countryman Janusz Gol.