The Cameroon striker scored in the 26th minute and 76th minutes with carbon-copy strikes after latching on to precise Wesley Sneijder passes.
Palermo, seeking their first Italian Cup success, set up a grandstand finale with Ezequiel Munoz's 88th-minute header but the Argentine defender was then sent off and substitute Diego Milito scored in stoppage-time to give the Nerazzurri their seventh success in the Coppa Italia.
Inter's Leonardo, winning his first trophy as a coach, was full of praise for his players after emerging from pitch-side celebrations more muted than those performed by AC Milan who clinched the title at the same Stadio Olimpico three weeks ago.
"It's been a difficult season with lots of injuries, three coaches and different systems of playing," said the Brazilian.
"They had many reasons to stop but they fought until the end. They deserved to end the season with a smile."
"Palermo started well. They played with lots of energy and really wanted to get an early goal," he said. "We found it difficult to get out, but we got our goals at just the right moments and were able to control the game."
Strike partner Javier Pastore, reportedly on Inter president Massimo Moratti's shopping list, then had two chances.
Inter struck with virtually their first attack as Palermo reorganised at the back. Sneijder released Eto'o with an inch-perfect pass and the Cameroon striker drew keeper Salvatore Sirigu before coolly poking home into the far corner.
The Inter goalkeeper then produced a fine save to tip the ball round the post as substitute Fabrizio Miccoli, on for Hernandez, headed Pastore's cross goalwards.
The Rosanero piled men forward but were caught on the break with a quarter of an hour to go as Snejider fed Eto'o who glided into the box before shooting calmly into the corner.
But the Argentine defender was immediately given his marching orders after a lunge on substitute McDonald Mariga and after Eto'o had hit the bar with the r
