The champions surged up to fourth and, with two games in hand, are just five points behind second-placed Napoli, who drew 0-0 with Fiorentina.
Dejan Stankovic, thriving in place of the injured Wesley Sneijder in Leonardo's new 4-3-1-2 formation, made the San Siro breakthrough after 20 minutes when he poked in Eto'o's pass.
"The pitch is not at its best and we made a few errors but it's all good," the Serb told Sky.
Striker Diego Milito, who hit the post with a header early on, netted his first league goal since November after 30 minutes before setting up Eto'o with a backheel just past the hour.
Camerooon striker Eto'o soon curled in a great freekick for the fourth before Henry Gimenez lashed in a consolation.
Inter's confidence has rocketed since former Milan boss Leonardo replaced Rafa Benitez on December 24 but the Brazilian's arrival has also coincided with the end of their injury crisis.
Captain Javier Zanetti, 37, was again inspirational in midfield as he equalled Beppe Bergomi's Inter record of 519 Serie A appearances, helping limit Bologna to just one first half chance when Luca Castellazzi saved from Marco Di Vaio.
Bologna, whose goalkeeper Emiliano Viviano has been linked with Inter given Julio Cesar's injury problems, were smarting from having two points deducted in midweek for payment problems.
Napoli, who beat Juve 3-0 last weekend thanks to Edinson Cavani's hat-trick, struggled for fluency at home to Fiorentina with midfielder Marek Hamsik banned and defender Gianluca Grava going off with a potentially serious first-half knee injury.
Fiore have had a disappointing season so far, made worse by striker Adrian Mutu being dropped from the squad after a row, but they bossed the game for long spells.
Walter Mazzarri's Napoli improved after the break and piled the pressure on at the end but a winner always looked unlikely in Serie A's first game of the second half of the season.
