Shola came off the bench to open the scoring with a deflected 84th-minute shot, and Sammy made sure in the fourth minute of stoppage time to book the Magpies' place in the third round of the Capital One Cup, something which had looked far from certain for much of the previous 80 minutes or so.
It was a first win of the season for manager Alan Pardew, who fielded a much-changed side, but was far from convincing.
A much-changed Cardiff side progressed in the League Cup with two second-half goals in the space of a minute eliminating plucky Accrington.
Aston Villa went some way to laying to rest the ghosts of last season's semi-final exit to Bradford by easing past League One Rotherham at Villa Park.
Paul Lambert branded that two-legged defeat to the then League Two club his lowest point as Villa boss, and he was not prepared to risk it happening again as he fielded a strong team which included Christian Benteke, Gabby Agbonlahor and Andreas Weimann.
And they were far too good for Steve Evans' side, with Lambert's selection justified by half-time as Weimann opened the scoring with a fine 20-yard effort in the 19th minute and then Benteke headed home Agbonlahor's cross 20 minutes later to ease the nerves.
Fabian Delph put the game to bed shortly after the interval with a fine team goal, latching on to Benteke's scooped pass to finish with aplomb.
The Belgian came off the bench to score in what could prove his final appearance for the Toffees at Goodison Park after 115 minutes of an absorbing second-round tie.
League One Stevenage had taken the lead against the run of play after 36 minutes through Luke Freeman but debutant Gerard Deulofeu levelled just before the break.
Fellaini had received a good reception on his introduction despite continually being linked with a move to Manchester United before next week's transfer deadline.
Team-mate Leighton Baines, who is also interesting United, was an unused substitute but was also applauded as he warmed up.
With the transfer window due to close before Everton's next home game, it remains to be seen whether either play in blue at Goodison again.
Fellaini's close-range strike sent the Merseysiders through to a third-round clash with Fulham.
Kenwyne Jones netted a hat-trick as Stoke saw off League One side Walsall 3-1 at the Britannia Stadium.
The Potters took the lead in the 22nd minute when the visitors squandered possession and Jones made the most of the opportunity, cutting inside and firing home.
The Trinidad and Tobago striker took only nine minutes to add another, rifling into the net from close range after the ball had dropped at his feet from a corner.
It was all fairly comfortable for the Potters, but Walsall did show some signs of promise in the first half.
And after Charlie Adam had been denied by the leg of Saddlers goalkeeper Richard O'Donnell early in the second period, the visitors got themselves back in the game through Ashley Hemmings' 57th-minute strike.
Stoke subsequently regained control of the contest, with both Andy Wilkinson and Robert Huth putting the ball in the net but having their efforts ruled out for offside.
And Jones then completed his treble in the 84th minute, racing on to a through-ball and finishing coolly.
Watford are through to the third round of the for the first time since 2008 after they beat Bournemouth 2-0 at Vicarage Road; while Jamaal Lascelles' extra-time winner saw Nottingham Forest beat Millwall at the City Ground.
Stoke City 3-1 Walsall
Aston Villa 3-0 Rotherham United
Morecambe 0-2 Newcastle United
Watford 2-0 AFC Bournemouth
Accrington Stanley 0-2 Cardiff City
Everton 2-1 Stevenage AET
Nottingham Forest 2-1 Millwall AET