Caballero guessed correctly to save Hazard's 35th-minute penalty but the Belgium global decisively dispatched the rebound - a goal that would preserve Chelsea's seven-point lead over Tottenham Hotspur and move them 14 above City with eight games to play.
If Chelsea do secure the title for the second time in three years, Hazard's contribution will be key.
Furthermore, Manchester United's frustrating times continued as they only managed to steal a point against Everton.
With each goal - 12 and counting after Wednesday - and every moment he's shimmering across the field, Hazard has rekindled the love affair with the Chelsea faithful. "It is not easy to defend 40 metres behind with Diego Costa, Hazard and these type of players and he did it really well". But I think we suffered as a team in the right way.
Guardiola, who felt City's performance was better than in Sunday's draw at Arsenal, said: "Absolutely (City were better than Chelsea)".
That's what it felt like on Thursday when Chelsea edged Manchester City 2-1 to keep their seven point lead over Spurs.
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The latest bit comes amidst the specter of the company having its shares delisted from the Tokyo Stock Exchange. Toshiba shareholders on Thursday agreed to split off its prized chip unit, paving the way for the sale.
Conte said: "A big win for us".
Swansea, who led through Wayne Routledge's early goal at the Liberty Stadium and seemed poised for a priceless victory, slipped back into the bottom three, two points below Hull. "So now we want to [get in] the Champions League places". That's the target. We have to be ready for everything.
"It's always a battle with him when we play one against one", the man-of-the-match told BT Sport.
It was a night to forget for the goalkeepers in west London.
A loose kick by home goalkeeper Thibaut Courtois - another Belgium global - enabled Sergio Aguero to level for City in the 26th minute.
Antonio Conte's men were in front again as Fernandinho fouled Pedro.
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Antonio Trillanes IV said corruption remained "rampant" in the government and called the firing of Sueno "a convenient excuse". There was no immediate indication of whether Sueno would also face criminal charges over the alleged offenses.
"If you stay in Tottenham, if you win it's great, great, but if you lose.it's not a disaster, no?"
The pair made rare starts in the midweek defeat to Chelsea, with both players completing their first full 90 minutes since the FA Cup win over Crystal Palace at the end of January.
And City in the wrong way. Now we've got those six points we virtually see it as a free hit.
The Italian is comfortable praising a coach who is not proving to be a direct threat this campaign.
If United win their games in hand, they'll go level on points with us, as will Arsenal.
Along with his familiar lament of missed chances, Guardiola also alluded to key decisions that have gone against his side when discussing the penalty incident. If we are able to do this, we will win the title.
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Nationally, distracted driving took the lives of 3,477 in 2015, a National Highway Traffic Safety Administration report states. In addition to the #77 program expansion, Porrino said an enforcement blitz has been initiated by the State Police.