The Ivory Coast striker headed in a left-wing corner from Frank Lampard [ Images ] in the 61st minute to reach 151 goals for the Stamford Bridge [ Images ] club, one more than Peter Osgood and Roy Bentley.
Only Lampard, who also netted on Saturday to take his tally to 182, Kerry Dixon (193) and Bobby Tambling (202) have scored more goals for Chelsea.
Drogba, 33, who joined the London [ Images ] club from Olympique Marseille [ Images ] in 2004, limped off with an injury late in the game against Bolton and is rated doubtful for the midweek friendly international against Guinea in Abidjan.
The Portuguese was widely criticised in midweek after he left the experienced trio out of his starting lineup as the side were beaten 3-1 by Napoli in their Champions League [ Images ] first knockout round first-leg tie in Italy [ Images ].
Chelsea have 46 points from 26 matches, three ahead of Arsenal and Newcastle United.
"The pressure has been on to get results recently," Villas-Boas said after his team claimed only their third win in 11 league games.
"Our run has not been impressive the last couple of games so the team wanted to go back to winning ways," he told the club website.
"This can mean a new start. It was important to win for the fans ... and it was important they could go home smiling."
David Luiz opened the scoring for Villas-Boas's men with a curling right-foot effort in the 48th minute and Lampard sealed victory by tucking the ball in at the far post from a right-wing cross with 11 minutes left.
Lampard's goal was the 150th of his league career.