The hosts made hard work of capitalising on the early dismissal of Nenad Krsticic, with Icardi having a penalty saved by Sergio Romero.
However, Shaqiri finally ended Sampdoria's stubborn resistance following a neat one-two with Podolski after 71 minutes.
The visitors then had Pawal Wszolek sent off soon after and Icardi wrapped up the victory three minutes from time when pouncing on a rare defensive slip.
"After the first sending off, Sampdoria got every man behind the ball and it proved to be harder than we expected," Icardi told Inter's official website.
"We didn't play at our best at the start, but then we made the breakthrough and got the win.
"Shaqiri and Podolski? They are two top-quality players. Hopefully we can go a long way with their help."
Icardi was relieved that his penalty miss did not prove costly after being outdone by his Argentina team-mate.
"I changed my mind at the last second. Romero knows me well and he managed to keep it out," added the 21-year-old. "Luckily I scored in the second half."
