Costa was an unused substitute as Jose Mourinho lost patience with the striker, but Chelsea's second away win of the season remained elusive as Tottenham extended their unbeaten run to 13 games.
Both sides had returned from lengthy European trips, with Spurs boss Mauricio Pochettino describing the noon kick-off as a "joke" following Thursday night's 2,500-mile flight from Azerbaijan.
Fluency was lacking in an at-times scrappy contest which lacked moments of real quality.
Hugo Lloris made a fine save to deny Eden Hazard a first goal of the season after Son Heung-min had spurned the best opportunity of the first half following Harry Kane's cross.
Mourinho will be grateful for a fourth Premier League clean sheet of the season, particularly in the absence of captain John Terry (ankle) and at a venue where Chelsea lost 5-3 on New Year's Day.
That defeat drew Manchester City level with the Blues at the top of the table, but Chelsea won the title at a canter with a fine second half of the season.
Mourinho's men are now 14 points behind leaders City and need an unprecedented run just to break into the European places.
Spurs' unbeaten streak, since the opening-day loss at Manchester United, is their longest since a 14-match run between November 1984 and March 1985.
Mourinho says he left out Costa due to low confidence after just four goals this season, but the pair argued in Israel and the manager criticised the striker's reading of the game on Friday.
Reports continue that Chelsea are looking for a new striker in January - with Kane among those linked - and the speculation is likely to heighten after Costa's omission.
Hazard, looking for his first goal of the season, was deployed as a "false nine", with Kurt Zouma replacing Terry, while the suspended Dele Alli was replaced by Ryan Mason for Spurs.
Chelsea's attacking quartet of Pedro, Willian, Oscar and Hazard caused a host of problems for Spurs with their movement and trickery, but clear-cut chances were hard to come by and Cesc Fabregas' guile was again curiously absent.
The best early chance was when Oscar crossed for Hazard, who headed just over.
Prior to that Tottenham had come closest to taking the lead.
Danny Rose had crossed towards Kane and Gary Cahill intervened, in the process catching Asmir Begovic on the head.
Begovic next saved Kane's shot, gathering at the second attempt before Hazard went close.
Pedro's effort from the edge of the box deflected over and Son lost Branislav Ivanovic to meet Kane's cross from the right, but headed weakly at Begovic.
Mousa Dembele next had a shot stopped and Son fired over, before Oscar hit the side-netting after beating Mason.
Mason went close to finding a way through the Chelsea defence only to be halted by Zouma as his ankle gave way
Erik Lamela replaced him and returned a pass for Son, who could only shoot at Begovic
Chelsea were under pressure which was so nearly relieved when Ivanovic's cross-field pass found Hazard, who struck a first-time volley across Lloris, who made a fine save.
That was as good as it got in a stalemate.
TWEET OF THE MATCH
"Looks like mourinho has come and parked the bus at the lane today" - former Tottenham defender Graham Roberts was not particularly impressed by Chelsea
Hugo Lloris: 7 (out of 10)
Kyle Walker: 6
Danny Rose: 6
Toby Alderweireld: 6
Jan Vertonghen: 6
Eric Dier: 6
Ryan Mason: 5
Mousa Dembele: 6
Harry Kane: 6
Christian Eriksen: 6
Son Heung-min: 6
Erik Lamela (for Mason, 56): 5
Clinton Njie (for Son, 74): 6
Asmir Begovic: 7
Branislav Ivanovic: 6
Cesar Azpilicueta: 6
Kurt Zouma: 6
Gary Cahill: 7
Cesc Fabregas: 5
Nemanja Matic: 7
Eden Hazard: 7
Kenedy (Willian, 89): 5
Ruben Loftus-Cheek (Pedro, 90): 5
Hugo Lloris: The France international made the crucial save from Eden Hazard in the second half to keep Chelsea out.
MOMENT OF THE MATCH
Lloris made a fine save to deny Hazard a first goal of the season
It was the one true moment of quality in the game.
VIEW FROM THE BENCH
Diego Costa was dropped after his half-time argument, with Pedro deployed as a false nine
John Terry was absent through injury, so Branislav Ivanovic returned at right-back and as captain
Dele Alli was suspended but Spurs had a first-choice line-up otherwise following their trip to Azerbaijan
The changes made little impact.
MOAN OF THE MATCH
Two top clubs looking to prove points to one another -it was a shame they could not serve up better than a drab stalemate.
WHO'S UP NEXT
West Brom v Tottenham (Barclays Premier League, Saturday, December 5)
Chelsea v Bournemouth (Barclays Premier League, Saturday, December 5)
Jose Mourinho was frustrated that Chelsea were unable to secure victory at White Hart Lane, although he was proud of his side's efforts.
"I think it was, as a team, the best Chelsea (performance of the season)," he said
"I think we deserved a bit more than what we got, but I am really happy with the performance.
"The team defended like a team, defended with 11 players
The team with the ball was comfortable against a team that presses really well and aggressively.
"It is difficult to play against them but we were comfortable with the ball and we had our chances, which is not easy to create against them.
"We had our chances and a couple of them were really, really big chances, especially the fantastic save Lloris did on Hazard.
"I am very happy with the performance
We are not happy with the result but we are happy with the team
I think we have the team again and we look to the future with a better spirit."
Asked about Diego Costa's place on the bench, Mourinho told BT Sport: "There is not an issue, not at all.
"It was just a decision and I think we played really well
The dynamic was very good.
"Hazard and Oscar appeared always in dangerous positions, the same as Pedro."
Source : PA