Pedro scored twice but it was Hazard, who netted with a penalty, who thrived in a central role as Costa and Kante served one-match suspensions and Chelsea won for a club-record 12th successive Premier League game.
"I think today we sent a good message," Conte said.
"A lot of people waited (to see if) without two really important players - Costa and Kante - we could lose points
It didn't happen."
Conte insisted he is not surprised by Hazard's ability and praised the Belgian's willingness to work for the team.
The Italian added: "I'm not surprised because he has real talent
The most important thing is he's putting his talent in the team."
Should Chelsea beat Stoke on New Year's Eve, the Blues can equal Arsenal's record 14-game winning sequence at Tottenham on January 4.
Conte is "proud" of Chelsea's run, which has seen them score 28 goals and concede two, while keeping 10 clean sheets, but is adamant the finishing position in the table is all that matters.
"If you ask me, I hope to continue until the end of this season," said Conte, with a smile.
"For me it's important only for the table
With these 12 wins in a row we took 36 points and our table is fantastic now.
"I always thought that the record is not important if at the end you don't reach something important in your championship.
"But to be with my players, to have this record, for sure we are proud about this.
"Now we want to continue in this way
For sure it won't be easy."
Michy Batshuayi was overlooked, but Conte says the £33million signing from Marseille will not be leaving Chelsea on loan next month.
Batshuayi came on only after Pedro's late third, facing only the restart before the full-time whistle, when he was consoled by Conte.
Batshuayi is yet to start in the Premier League, but Conte says it would be a "defeat" for him as head coach if the striker is permitted to make a temporary move.
Conte repeatedly said "no" when asked if Batshuayi would be allowed to go out on loan.
"This player is a new player for Chelsea and to go on loan now is a defeat for the club and above all for me," Conte said.
"We have to continue to work with him every day to try together to improve him.
"I want to win and every decision I make is because I want to win and I have to take the best decision.
"Michy Batshuayi is a really good player, but he's a young player and he's trying to adapt to this league, this football.
"I try to make the best decision for the team, but I'm sure Michy could be the possibility to show his talent in the future."
Bournemouth boss Eddie Howe felt Chelsea's second, when Simon Francis felled Hazard, was the key moment of the game.
Howe said: "The decisive moment was the second goal and how that went against us I thought was a key moment."
Asked if he felt aggrieved at the penalty award, Howe added: "I don't know
I haven't seen it again and it was a difficult view for me
I didn't think it was contentious at the time."
Howe admitted Bournemouth, who won this fixture last season, struggled with Chelsea's three-pronged strike-force, particularly Hazard.
"I thought he was outstanding today, there's no doubt about that," the Cherries boss said.
"His movement, his ability to turn on the ball so quickly
His technical ability, very high level.
"When you have him and Pedro and Willian counter-attacking on you at such pace it's difficult.
"That was the thing where we fell short today
On the ball we gave them too many times to get in those situations, by not being tight enough on the ball and good enough on the ball in certain situations."
Source : PA
Source: PA