The Foxes gained the lead in the 11th minute after a deft Wes Morgan flick crept into the net off the post in what was the defender's second goal in as many games.
Bournemouth nearly levelled through Callum Wilson, whose shot was saved superbly by Kasper Schmeichel.
Leicester continued where they left off in the second half and clinched the game in the 82nd minute after a Youri Tielemans cross was converted by Jamie Vardy's head via the underside of the crossbar.
It ended 2-0 to Leicester as the home side paid tribute to late chairman Vichai Srivaddhanaprabha ahead of his birthday.
'I thought we were excellent'
Speaking to BBC Sport after the match, Rodgers reflected on his side's winning performance.
He said: "I thought we were excellent. We showed a real appetite and hunger and tactically we were very good.
"Our pressing was excellent, the boys are understanding tactically how we want to press and the importance of that."
He added: "Each game, each day in training, we are getting more used to it."
'That’s why we’re here, we’re here to win games'
Rodgers spoke further after the game to Leicestershire Live, and praised Foxes' fans for their impact on the game. He said: "Sometimes supporters can get criticised for being quiet or whatnot, but the players on the pitch, they make the supporters happy.
"That’s why we’re here, we’re here to win games, play a brand of football that’s aggressive and attacking and I think the supporters responded to that second half."
He added: “They could see what the players were giving . On a day that was a special occasion they showed their appreciation, which was a lovely touch.
“That generated through to the players and gave them energy, which is vital for any team.”