Vardy, who made his England debut in the friendly against the Republic of Ireland in June, won and converted a spot-kick in the 88th minute.
Eddie Howe’s side appeared to be on course to seal their first home win in the Premier League following Callum Wilson’s acrobatic strike midway through the first half.
But with only four minutes remaining, Vardy was upended by Steve Cook in the penalty area.
“I am very pleased because it was a tough match,” said Ranieri as Leicester retained their unbeaten start to the season.
“Bournemouth are very well organised in every moment and after we conceded a goal we lost our composure.
“The manager (Howe) is doing very well, and I told him at he beginning of the match, ’well done’ because they play like an orchestra.”
Howe was left to rue knee injuries to Tyrone Mings – the £8million summer arrival from Ipswich – and Max Gradel.
Mings was left writhing in agony after a collision with Leicester’s Danny Drinkwater just five minutes into his debut.
The 22-year-old defender immediately signalled to the bench that he could not continue and received gas and air before he was carried off on a stretcher.
Max Gradel’s afternoon also ended prematurely following what appeared to be an innocuous challenge from Ritchie de Laet, while Charlie Daniels was withdrawn at the interval with a knee injury.
“It was difficult for us, second half especially with the injuries,” said Howe.
“It restricted us tactically — we couldn’t make any changes late in the game when we needed to get momentum back in our play — and also from the guys’ perspective to see their team-mate stretchered off.
“They are so tight here it is a difficult one psychologically to handle.”
“Tyrone didn’t look good and seeing him in that much pain isn’t nice to see, and Max also knew pretty early that he had a serious one there and any knee injury is worrying, so concerns for all three players tonight.”
The Bournemouth boss also claimed his side should have been given the opportunity to double their lead.
That incident occurred when Lee Tomlin appeared to be tripped in the penalty area. Referee Neil Swarbrick awarded only a free-kick.
“I feel very aggrieved over Lee Tomlin’s penalty which is clearly in the box,” said a clearly annoyed Howe afterward.
Boruc, Francis, Elphick, Cook, Daniels (Mings 45), Ritchie, O’Kane, Surman, Gradel (Pugh 67), Tomlin, Wilson
Schmeichel, de Laet (Benalouane 91), Huth, Morgan, Schlupp, Drinkwater, King, Mahrez (Okazaki 45), Kanté, Albrighton (Dodoo 71), Vardy