The Brazilian gave the Blues the lead against the run of play in the first half before setting up the Spaniard to notch his 15th Premier League goal of the season
Having failed to claim victory in their previous two Premier League outings, Jose Mourinho's title hopefuls were not at their best on Saturday but coped well with Newcastle's threat.
Ultimately, they struck at key times either side of the break to go two points clear of Manchester City at the top of the Premier League, following the champions' 1-1 draw with Everton.
The hosts were on the back foot for long spells of the first half but led at the break when Oscar converted Branislav Ivanovic's cross from close range.
There was very little that fit-again Newcastle goalkeeper Tim Krul could do about that strike on his return from an ankle injury, and the Dutchman was similarly helpless when Costa made it two just before the hour mark.
John Carver once again led Newcastle following Alan Pardew's recent departure, and any long-term replacement in the dugout will likely take positives from their performance, even after a fifth straight Premier League away game without a win.
Despite having endured a difficult few weeks with Pardew's exit and the FA Cup defeat to Leicester City, Newcastle started well, with Moussa Sissoko's effort on target and Remy Cabella looking lively early on.
Cabella was part of four changes from the 3-3 draw with Burnley last time out in the Premier League and the Frenchman went close with a curled effort 19 minutes in that deflected wide off John Terry.
However, Chelsea offered a reminder of their quality on the break, with Jack Colback required to make a timely interception.
Oscar fired wide from inside the area when he perhaps should have done better, although it was Newcastle who continued to threaten - Cabella cutting in from the right and forcing a smart save from Petr Cech, preferred to Thibaut Courtois in the Chelsea goal.
Sissoko struck the outside of Cech's left-hand post after the half-hour mark, with Cesar Azpilicueta’s injury-enforced withdrawal seemingly adding to Mourinho's frustrations before the end of the half.
However, slack defending cost Newcastle dear just before half-time as Willian found an unmarked Ivanovic from a quick corner and the Serb's delivery reached Oscar, who made no mistake with the finish.
Costa should have doubled Chelsea's lead early in the second half but he fired over from inside the area, with the Spain international livid soon afterwards as referee Roger East waved away his penalty appeals following an apparent handball from Fabricio Coloccini.
Mourinho's men were enjoying their best spell of the game and Costa soon doubled the advantage after 59 minutes by firing past Krul following a clever flick-on from Oscar inside the area.
As Newcastle laboured in an attempt to get back into the game, Costa was then denied a second by Coloccini's clearance, but in the end Mourinho's side claimed their first points of 2015 with relative ease.