Mauro Icardi got the ball rolling for the Nerazzurri by scoring in the 13th minute and Samp was made to regret Maxi Lopez's missed penalty when Walter Samuel doubled midway through the second half.
Icardi came back to haunt his old club for the second time only seconds after his fellow Argentine's strike and Rodrigo Palacio completed the victory with a late fourth.
Icardi gave Inter the early lead when he volleyed Palacio's cross past the helpless Angelo Da Costa.
The striker's decision to cup an ear at home fans saw him booked, but Sampdoria missed a golden chance to draw level from the penalty spot.
The spot kick – conceded when Andrea Ranocchia brought down Daniele Gastaldello – was well saved, with Handanovic moving to his right and saving to keep his side in front.
Sampdoria was dealt a further blow with Eder's dismissal, as the Brazilian was adjudged to have dived following a Samuel challenge.
The aftermath was messy as several players tangled, but it was Eder's contact with the face of Handanovic that saw him given his marching orders in the 20th minute.
Sampdoria kept pressing, as Handanovic kept out a curling Lopez effort and Gianluca Sansone's free kick, and the goalkeeper's stellar form continued as he denied Roberto Soriano just before the interval.
Handanovic's work would prove crucial as Samuel effectively sealed Inter's win 15 minutes into the second half.
The Argentina defender headed in Hernanes' corner and just three minutes later, Icardi kept cool to roll the ball home from a tight angle.
There was no big celebration on this occasion, but the game was already won, with Palacio capping the fine performance with a composed finish 11 minutes from time.
Palacio played a neat one-two with Ricky Alvarez and then strode unopposed into the Sampdoria penalty area, before slotting past Da Costa, as Inter registered its first win in five league matches, keeping it on course for a Europa League berth.