Ex-Crystal Palace and Sunderland forward Phillips, 40, netted eight minutes after coming on as a 72nd-minute substitute to send the Foxes 10 points clear at the head of the Sky Bet Championship.
Phillips headed an Anthony Knockaert cross past a diving Lee Camp and wheeled away in celebration but the home side were sure the whole of the ball did not cross the line.
Referee Darren Drysdale had no second thoughts though and the strike by former England player Phillips, who has already won four promotions in his career, makes it a club record nine wins in a row for Leicester.
Pearson said: "Kevin is a striker and is smiling, he knows he has scored. I have not had a clear look at the incident and the goal but the assistant seemed pretty sure it went over the line.
"I know that he will contribute very positively to our season both in terms of what he can do on the pitch and off the field as well. He has a lot of experience.
"He is a goalscorer and his anticipation is excellent, he is in his forties so won't be as quick as he was 16 years ago but still creates a lot of space and time for himself because of his understanding of the game.
"I couldn't imagine we would be 10 points clear. If you make assumptions then you are letting yourselves in for possible disappointment so we must be professional."
Before Phillips entered the field the league leaders had been denied by a superb string of saves from an inspired Camp.
The ex-Derby stopper denied Lloyd Dyer after 17 minutes before diverting a Jamie Vardy effort onto the bar two minutes later.
Vardy then had a shot hacked off the line by Tommy Elphick after Camp had closed him down quickly in a one-on-one situation.
Before Phillips netted the winner Camp superbly kept out a Knockaert header but it was not enough to win a valuable home point.
Bournemouth manager Eddie Howe said: "Our goalkeeper has had an excellent day between the sticks. He kept us in the game and we thought that might get us a point but it wasn't to be.
"The feeling from my players is that it wasn't a goal. I have only had a quick look at it and it is very close, you cannot really tell as it is a really bad angle."
"They showed what a good side they are. Nine wins in a row is a remarkable achievement and credit to them."