Bak caught his rivals napping as he launched his decisive move in the pretty Mediterranean costal town of Sestri Levante at the conclusion of the undulating 155km stage through the Ligurian hills.
Spaniard Joaquim Rodriguez (Katusha) retained the leader's pink jersey by 17 seconds over Canada's Ryder Hesjedal (Garmin-Barracuda), although Casar moved into third place after the peloton rolled home 3:34 in arrears.
Played out under bright sun and blue skies, Thursday's stage started in the coastal town of Seravezza at the foot of the Apuanian Alps.
After 40km of fast riding along the east coast of north Italy, a nine-man group formed ahead of a succession of classified climbs in the final two thirds of an intriguing stage.
Polish youngster Michal Golas (Omega Pharma-Quick Step) crossed the first three of four climbs in pole position to take enough points in the mountain classification to wrest the blue jersey from the shoulders of stage six winner Miguel Rubiano (Androni Giocattoli).
After attacking towards the summit of the Cat.2 Valico La Mola 33km from the finish, Golas opened up a small gap over his fellow escapees. Despite overcooking three bends on the subsequent descent, Golas managed to increase his lead to around 25 seconds at the foot of the final climb of the day, the Cat.3 Villa Tassani.
Further back, the peloton seemed content to let the break run its course: a maximum gap of seven minutes put Casar - who trailed Rodriguez by 4:01 in the overnight standings - in the virtual maglia rosa.
The presence of Italian Ivan Santaromita (BMC) in the group - himself a further 16 seconds back in the GC - added another dynamic, and with time bonuses available at the finish, both men had realistic designs on the fabled pink jersey.
The Katusha team of Rodriguez nevertheless refused to lead the chase, leaving the responsibility to the Liquigas team of former winner Ivan Basso and the Lampre squad of defending champion Michele Scarponi.
An exhausted Golas was caught by the chasing group on the steep 11% section at the start of the final climb before Casar responded to an attack by Amador to take the points over the summit ahead of Santaromita.
The trio began the final descent in the lead but soon the race came back together on the flat approach to Sestri Levante. With the peloton crossing the summit almost four minutes adrift, both Casar and Santaromita still harboured hopes of ending the day in pink.
But with the pace slowing as the leaders started to play games ahead of the finish, Bak seized his chance and powered ahead.
There was no response from the remainder of the break, and three-time national time trial champion Bak used his strongest weapon to stay out ahead. The rangy veteran had time to savour the moment, blowing kisses to the crowd and pointing to his sponsor as he crossed the line in a time of three hours, 58 minutes and 55 seconds.
Casar survived a late surge from Amador to take second place - and with it some potentially vital bonus seconds - ahead of the Costa Rican, Jan Bakelants (RadioShack-Nissan), Santaromita, Jackson Rodriguez (Androni) and Amets Txurruka (Euskaltel).
Team Sky's Juan Antonio Flecha led the peloton over the line 3:34 in arrears, ending Casar's short-lived dream of taking the maglia rosa.
The Giro continues on Friday with the short 121km stage 13 from Savona to Cervere, which features just one classified climb but a series of hills as the race edges closer to the Alps.
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