For the second consecutive week, it was Chelsea's persistence - not their possession - that proved decisive.
Until that point, Bournemouth had been more consistently unsafe. They could also have been behind, had Wilson and Ake been more clinical.
Alonso picked up the ball on the left-hand side before threading a pass through to Hazard, who wasted little time before drilling a low effort into the far corner of Begovic's net.
Sarri is delighted with the win and is happy that his players showed the patience to keep playing their football in order to get the goal as he was confident that breakthrough would come in the final quarter of the game.
Maurizio Sarri has hailed Marcos Alonso as the best left-back in Europe and believes the Chelsea defender will be regarded as the best in the world if he improves. Chelsea though, are nearly at full strength. Luckily for Chelsea, he could only divert Sergio Rico's pinpoint cross wide. Andreas Christensen will have to make do with a place on the bench, with Gary Cahill still off the pace.
Fans had always quietly considered him an underrated player and his performances in a Chelsea shirt are really showing off his quality.
'He's very young for a striker, and can improve more.
Stamford Bridge has been a happy hunting ground for Bournemouth - they had won two of their previous Premier League visits and triumphed 3-0 last season - but as Howe made clear prior to kick-off, Chelsea present a very different challenge under Sarri.
And having struggled latterly during his time in Spain, quickly adapting to the Premier League was never going to be an easy task. "Because the gap at the beginning was 30 points", Sarri said.