Tottenham had seen off Watford 4-0 earlier in the day to cut the gap to four, but Antonio Conte’s men sealed three points at the Vitality Stadium to maintain their push for a fifth league title.
Adam Smith gave Chelsea the advantage in the 17th minute as he turned the ball into his own net from a Diego Costa shot, before Eden Hazard superbly slotted home the second not long after.
Josh King netted his tenth goal in 11 Premier League games to pull a goal back for the Cherries just before half-time, but Marcos Alonso sealed the three points for the Blues with a sublime free-kick in the 68th minute.
Bournemouth, who are seven points ahead of the relegation places, made one change from the midweek draw with Liverpool while Conte welcomed Victor Moses and Nemanja Matic back into the starting XI.
It was an even start, but Chelsea took the advantage in the 17th minute. A ball forward from David Luiz picked out Moses on the right and his cross reached Diego Costa, who sliced his attempted shot. But it nestled into the back of the net after being diverted goalwards by the body of Smith.
Just three minutes later, the Blues added another. N’Golo Kante, with his first Premier League assist for the club, picked out Hazard on the left and he danced towards the box, swerving around the oncoming Artur Boruc to coolly slot home.
Bournemouth could have pulled a goal back after half an hour as Charlie Daniels fired into the box a ball that found Benik Afobe on the right. The striker hit a fine volley, but it could only rattle the crossbar and nick the foot of Thibaut Courtois before going wide.
While Chelsea kept pressing for another goal – Moses just heading wide in the 38th minute – the hosts did pull a goal back with three minutes of normal time to play.
Afobe found King on the right and the in-form forward curled home although the ball did take a nick off the thigh of Luiz on the way through.
It could have been all square on the 45-minute mark as King and Afobe linked up one again, but Cesar Azpilicueta got a vital flick onto a cross from the former to see the ball evade the head of the waiting Afobe and go wide.
The second period was a quieter affair although the attacking intent was still there from both sides, with the first real chance of the half coming after 57 minute.
Costa found Alonso on the left and the wing-back took one touch before passing back into the box for his team-mate. But he could not get a toe to the cross to poke home and the home defence cleared.
Substitute Issy Mousset could have made things interesting on the 90-minute mark when a Daniels cross found him in the box, but Gary Cahill was able to nick the ball out of play before he could shoot and Chelsea saw out the victory to end Bournemouth’s run of five Premier League games unbeaten.
