A first-half goal from Aaron Lennon, the first time since August 2009 he has scored in back-to-back games, and two late Ross Barkley penalties secured a 3-0 victory against a disparate visiting side who lacked cohesion or any semblance of a game plan and who defended haphazardly at best.
Martinez has now won 40 times and drawn 31 in his opening century of games at Everton but after a Capital One Cup semi-final exit last week some fans were calling for his head.
A victory and a much-needed home clean sheet, only the second of the season, will have eased that clamour.
McClaren's record is even worse, having taken just five points from eight matches since their last victory on the road at Tottenham in mid-December, and after spending £24million on Jonjo Shelvey and Andros Townsend in the January window he needs to see a return on his investment pretty quickly with his side two points from safety.
The match followed a familiar pattern for Everton - who were without the injured defender John Stones - at Goodison Park this season with the scoreline not really reflecting their domination of possession.
Lennon took his goal well but could easily have had another two either side of the break.
Tom Cleverley forced Rob Elliot into one early low save, Romelu Lukaku flicked a free-kick just past the post and Barkley, on his 100th Premier League appearance, lobbed wide of an open goal after Elliot had rushed out to head clear under pressure from Lukaku.
The breakthrough came in the 23rd minute when Lennon was picked out by Cleverley and the winger's shot on the turn beat Elliot.
It looked like being the only thing which would do so all night as the Magpies goalkeeper produced a string of saves in the second half, which Everton played without leading goalscorer Lukaku who he was replaced by Arouna Kone after struggling with a back injury sustained in a challenge with Fabricio Coloccini.
Barkley had two shots, one a free-kick deflected by Shelvey, tipped onto the crossbar by Elliot, who also saved from Lennon from which Kone fired the rebound wide.
The post denied Cleverley from a set-piece as Everton continued to create but not take chances.
It is one of the reasons Everton made a deadline-day move for Oumar Niasse, restricted to a watching brief in the stand still wearing the club shirt in which he was presented to the crowd in pre-match, who scored 13 goals in 23 appearances for Lokomotiv Moscow.
Unfortunately for Newcastle Ivory Coast striker Seydou Doumbia, who received his visa earlier in the day after his loan move from Roma, was forced to do the same from the bench as bad luck with injuries saw the visitors make all three substitutions with just 55 minutes gone.
Newcastle did give a debut to Townsend, named in the starting line-up, but after so long in the wilderness at Tottenham he was an anonymous figure.
Moussa Sissoko finally forced Joel Robles, making his first league start for a year after injury to Tim Howard, into a save in the last 15 minutes but just seven goals in their last 18 league matches away from provides some explanation why they remain rooted in the bottom three.
Barkley made the result safe with an 88th-minute penalty after substitute Rolando Aarons brought down Lennon, before a cheeky chip from the spot capped an impressive performance after Newcastle substitute Jamaal Lascelles was sent off for bringing him down.
TWEET OF THE MATCH
"Shocking #nufc" - Newcastle Evening Chronicle chief sports writer Lee Ryder can offer readers at home little encouragement with his succinct summary of the Magpies' performance.
PLAYER RATINGS
Everton
Joel Robles: 7
Seamus Coleman: 7
Phil Jagielka: 6
Ramiro Funes Mori: 6
Bryan Oviedo: 6
Gareth Barry: 6
Aaron Lennon: 7
James McCarthy: 6
Ross Barkley: 7
Tom Cleverley: 7
Romelu Lukaku: 5
Substitutes
Arouna Kone (on for Lukaku, 46): 5
Newcastle
Rob Elliott: 8
Daryl Janmaat: 6
Fabricio Colocinni: 6
Chancel Mbemba: 5
Paul Dummett: 4
Jonjo Shelvey: 5
Henri Saivet: 5
Andros Townsend: 5
Georginio Wijnaldum: 5
Moussa Sissoko: 5
Ayoze Perez: 5
Substitutes
Jamaal Lascelles (on for Mbemba, 44): 3
Rolando Aarons (on for Dummett, 46): 4
Aleksandar Mitrovic (on for Saivet, 55): 4
STAR MAN
Rob Elliot
The Newcastle goalkeeper was beaten three times, although two were penalties, but without him the scoreline would have been much greater after a string of saves after half-time.
MOMENT OF THE MATCH
Ross Barkley's cheeky chipped added-time penalty
Safe in the knowledge the game was already won the England international floated his second spot-kick down the middle of the goal.
VIEW FROM THE BENCH
Roberto Martinez made only one enforced substitution with striker Romelu Lukaku struggling with a back injury and was rewarded with only a second home clean sheet of the season
Newcastle boss Steve McClaren spent plenty of time in the technical area but failed to have any effect on his hapless side.
MOAN OF THE MATCH
Newcastle's tactics were not only negative but aimless as Everton, who have the league's worst home defensive record, were allowed to have as much as the ball as they wanted and some of the Magpies' late tackling was mindless.
WHO'S UP NEXT?
Stoke v Everton (Premier League, Saturday February 6)
Newcastle v West Brom (Premier League, Saturday February 6)
Source : PA
Source: PA