David Nugent and Jamie Vardy scored a goal in each half to keep Leicester's fine home run going.
Nugent's clinical finish brought him an eighth goal of the season in the 17th minute, while Vardy raced through and rounded goalkeeper Ryan Allsop to restore the Foxes' lead in the 64th minute.
In between times Marc Pugh, who grabbed the Cherries' dramatic equaliser against Nottingham Forest last week, had levelled matters with a spectacular 20-yard strike.
But Leicester dominated the second half and Bournemouth's cause was not helped when Elliott Ward was sent off for a second bookable offence in the 66th minute, after fouling Danny Drinkwater.
The victory was Leicester's ninth of the season and their sixth consecutive league win at home.
Pearson said: "I think we deserved to win. It was a decent performance against a good side who didn't surprise us.
"They set themselves up in a similar fashion as they played against Forest.
"They are a very decent side so for us to come out on top shows that we continue to progress. I thought they came with a positive intent and started the game well.
"I don't think it was a case of them sitting back because they are comfortable in possession and when they have the ball they use it pretty well.
"But I was really pleased with how we went about the game as well. We always looked dangerous in wide areas and have strikers who have good energy levels.
"It was good for both strikers to get on the scoresheet.
"Nugent went off in the second half because he was feeling ill, and I thought Chris Wood did really well and had a positive impact when he came on as substitute.
"It is all about winning games whether home or away and I thought it was a good performance. It was a good win for us."
Cherries boss Eddie Howe insisted his players could take great encouragement from their performance and had no complaints about Ward's red card.
Howe said: "I thought in the first half we were well in the game and had some really nice passages of play and a couple of good opportunities where we didn't really work the goalkeeper.
"We conceded a goal and showed good character to not let that affect us and scored a great goal ourselves.
"At half-time you think it could go either way.
"But in the second half I don't think we quite hit the level we did in the first half and it was backs to the wall.
"We were coping okay but conceded a poor goal and the sending off changes the game from our perspective. It was very difficult to build any pressure.
"We felt we had weathered the storm, defended set plays very well and hung in there.
"We felt that the longer we could keep it at 1-1 we would have a great chance to win the game.
"But that spell with the goal and the sending-off knocked us because you struggle to keep the ball with 10 men. I have no complaints about the sending-off. I think they were both yellow cards.
"But I think we can take encouragement from our performances in the two games against Forest and Leicester."