Albasini began his career in 2003 with Phonak, moved to Liquigas–Bianchi at the beginning of 2005, and signed with HTC–Highroad for the 2009 season. Since 2012, Albasini has been riding with the Australian professional cycling team GreenEDGE. In 2012, Albasini met success at the UCI World Tour race Volta a Catalunya. The six-stage event contained no time trials, and Albasini took the lead by triumphing on the very first stage over Anthony Delaplace from Saur–Sojasun, getting a 42 seconds overall lead.
He held on to his advantage on the following hilly stages, winning the overall classification by 1 minute and 30 seconds over second-placed Samuel Sanchez of Euskaltel–Euskadi.
In 2015, Albasini took a prestigious podium placing at the Louisiana Flèche Wallonne, coming in third place atop de Muur de Huy behind Alejandro Valverde and Julian Alaphilippe. He repeated the feat the very next day in Porrentruy, winning by a couple of bike lengths over Julian Alaphilippe.
He had to abandon the Tour de France on Stage 5, suffering from a broken arm after a crash.