Chelsea’s triumph, which moves them up to 12th, was soured by the sight of Terry limping out of the action as a Tuesday trip to Paris Saint-Germain in the Champions League lies in wait.
If Costa’s appearance was something new, the sight of him sniffing out a chance and sliding home Willian’s pass on the angle prompted familiar celebrations in the fifth minute.
Four minutes later, the visitors had a mountain to climb when – following a poorly delivered free-kick from Jonjo Shelvey – makeshift left-back Rolando Aarons played a woeful pass inside under pressure from Costa to Steven Taylor.
Pedro pounced mercilessly and raced towards goal to send a clinical finish beyond Rob Elliot from outside the penalty area.
There was worse to come for Steve McClaren’s beleaguered side – an unchanged line-up playing like strangers – as Costa powerfully outpaced Fabricio Coloccini and returned Willian’s favour from earlier by playing a diagonal ball across the box for the Brazilian to make it three.
Pedro shot too close to Elliot in the 37th minute as another presentable chance arrived courtesy of a Newcastle error – Shelvey the guilty party on this occasion.
Terry was substituted after landing awkwardly in a challenge with Aleksandar Mitrovic, marking the only blemish on the opening 45 minutes for Hiddink.
McClaren responded to his team’s hopeless predicament by replacing top scorer Georginio Wijnaldum with Jack Colback in midfield and they at least began the second period on the front foot, as Cesar Azpilicueta blocked a Shelvey shot and Mitrovic whistled an attempt past the far post.
The respite would not last, though, with Fabregas allowed overly generous time and space to float an eye-catching pass over Taylor for Pedro to net his second.
Costa then made way to a standing ovation for Traore, in order to avoid a repeat of Terry’s fate as the French and English champions prepare to face one another.
Newcastle’s abject showing – which featured the final blow of Traore steering in Azpilicueta’s 83rd-minute cross at the near post – leaves them contemplating a season facing far less-illustrious opposition next term.
Indeed, it will take much more than just Towsend’s splendid late long-range effort that beat Thibaut Courtois to lift the misery.