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.
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."