Chelsea, chasing their first league title for 50 years, also gained ground on third-placed Manchester United, who were held to a highly controversial 0-0 home draw by Tottenham Hotspur.
United keeper Roy Carroll fumbled a 90th minute long-range shot from Spurs midfielder Pedro Mendes, television pictures showed the ball had crossed the line by about a metre but Carroll scooped the ball back out and no goal was awarded.
Everton beat Portsmouth 2-1 and Bolton Wanderers registered their first win since October by beating Birmingham City by the same scoreline, with both games decided in the 90th minute.
Chelsea's fifth consecutive win since the week before Christmas left them top on 55 points, followed by Arsenal (48), United (44) and Everton (43).
Drogba, the club's record signing when he joined from Olympique Marseille for 24 million pounds ($46.04 million) in July, struck twice in the opening 17 minutes to move Chelsea closer to glory in their centenary year.
The Ivory Coast striker muscled past Gareth Southgate to score his first in the 15th minute and then beat Boro's experienced central defender again in the air two minutes later to head home a Frank Lampard free kick.
CHELSEA PRESSURE
Chelsea kept pressing for a third after the break but the match had already been won and the damage done to the title hopes of both their immediate rivals.
Chelsea assistant manager Steve Clarke told Sky Sports News: "We're seven points clear but we're only focused on what we're doing.
"We knew that over the Christmas and new year period we needed 12 points and we knew that would put pressure on other teams -- that's what we've done."
Over at Highbury, Manchester City made the breakthrough after 31 minutes with a superb shot from 25 metres by winger Shaun Wright-Phillips, watched from the stands by his stepfather Ian Wright, Arsenal's all-time top scorer.
Sweden midfielder Fredrik Ljungberg spared Arsenal's blushes, and a first home league defeat since May 2003, by heading an equaliser with 15 minutes left.
"Overall, I feel unlucky not to have won this game," Arsenal manager Arsene Wenger said. "But Man City played well and gave everything...sometimes you have a draw like that."
United fared no better than Arsenal in their pursuit of Chelsea with a draw that immediately sparked fresh calls for technology to be used in football.
Alex Ferguson's side dominated possession and created chances but were effectively blocked by a well-disciplined Spurs side who have now won five and drawn two of their last seven games under new Dutch boss Martin Jol.
However, the hosts only avoided almost certain defeat when match officials failed to spot the consequences of Carroll's fumble.
Shown a reply of the incident, Jol said: "This is amazing...I'm pleased with the point but we should have had three.
"I could understand it ...if it was maybe a couple of centimetres (over the line), but this is two foot...you can understand it, but it's still a disgrace."
Ferguson backed Jol's call for more technology in the game, saying: "It could have won the game for them."
Everton, playing hours after the six-million-pound capture of striker James Beattie from Southampton, ended a run of two consecutive defeats with their late win at Goodison Park, courtesy of Leon Osman's last-minute strike.
Kevin Nolan did the same for Bolton as Sam Allardyce's men finally returned to their winning ways at Birmingham.
