Premier League: Chelsea 3 Tottenham 0
Spain forward Costa - scorer of 11 goals in 11 Premier League matches this season - missed Wednesday's London derby at Stamford Bridge due to suspension, but that mattered little.
Chelsea legend Drogba, who returned to the club from Galatasaray in July, led the line superbly as Jose Mourinho's unbeaten side - who top the table by six points - overcame a bright start from the visitors to win comfortably.
Harry Kane hit the crossbar early on for Tottenham, but Chelsea responded like the champions many fancy them to become with two quick-fire goals.
Eden Hazard set Chelsea on their way in the 19th minute before Drogba netted less than 180 seconds later for his second league goal of the season.
Substitute Loic Remy, who replaced Drogba, sealed the convincing home win 17 minutes from time as Tottenham's winless run at Chelsea - which dates back to 1990 - continued.
Despite the absence of Costa, Chelsea fielded a familiar and settled line-up, while Tottenham, fresh from six wins in their last seven matches in all competitions, made just one change - with Roberto Soldado dropping to the bench despite his rare goal in a 2-1 win over Everton on Sunday.
Erik Lamela replaced Soldado for Spurs, who started brightly, and only the woodwork denied Kane in the seventh minute.
Aaron Lennon, continuing to impress after a recent recall, crossed for Kane from the right and the striker peeled off Gary Cahill to steer a header onto the bar, with goalkeeper Thibaut Courtois well beaten.
Kane then stripped Cahill off the ball in the 12th minute and charged into the box, only shooting just wide with his left foot.
The early chances buoyed the vocal travelling supporters, but they were soon silenced by a simple but brilliant Chelsea move.
Hazard played a neat one-two with Drogba and the Belgian glided past three Tottenham players before shooting under Lloris from a tight angle.
And Lloris was to blame for Chelsea's second goal - scored just three minutes later - after his sloppy clearance fell to Hazard.
He fed Oscar and the Brazilian's excellent ball played in Drogba, who beat Jan Vertonghen and took two touches before slotting home.
Chelsea dropped the tempo slightly following Drogba's strike, happy with their two-goal lead, but Willian shot twice at Lloris as the first half winded down.
Cahill took a knock to the head in the early stages, and was replaced at the break by highly rated Kurt Zouma, who had work to do early as Chelsea sat back.
The best Spurs could muster, though, was a tame Christian Eriksen shot from distance, and a wayward Vertonghen header, before Drogba came off to a healthy ovation.
And his replacement, Remy, wrapped up the three points, beating Vertonghen with strength before side-footing past Lloris in stylish fashion as Chelsea underlined exactly why they are widely considered title favourites with a dominant display.