The Spaniard opened the scoring against the Cherries but later received a booking to rule him out of the match.
Pedro tripped Brad Smith and referee Mike Jones had little option but to brandish the yellow card.
The 29-year-old had given Chelsea the lead with a fine effort to beat Artur Boruc in the Bournemouth goal.
And Eden Hazard converted a penalty in the second half to all but secure the points.
That means Chelsea will likely extend their lead over Liverpool at the top of the Premier League to nine points.
The Reds face Stoke tomorrow at Anfield.