Carver, without a win in four games in caretaker-charge, watched Frenchman Remy Cabella open his Magpies account after pouncing on an Andy Robertson error 10 minutes before half-time.
Ameobi's long-range second and a late strike from substitute Gouffran confirmed a third straight defeat for the hosts and plunged them deeper in the relegation mire
Hull were the architects of their own demise with occasional bright spells of play undone by two mis-placed passes which led directly to the first two Newcastle goals.
And they also contributed a whiff of controversy after Ahmed Elmohamady blatantly punched a Gaston Ramirez free-kick into the net in first-half injury time, only to be booked for his troubles after his indiscretion was spotted by an eagle-eyed assistant.
Newcastle had started with Papiss Cisse on the bench after his belated return from African Nations Cup duty and he might have needed extra time to acclimatise as the temperature plunged towards freezing.
The home side looked more vibrant in the early stages and almost grabbed the lead in the eighth minute when Michael Dawson headed just over from a Ramirez free-kick
Somehow the visitors stayed on level terms 10 minutes later after a mighty goalmouth scramble sparked by an Elmohamady cross from the right.
Tigers captain Curtis Davies headed the ball goalwards but a combination of Tim Krul's hand and the chest of Fabricio Coloccini managed to bundle the ball off the line.
Robertson fired a tame effort at Krul before Newcastle began to get to grips with their surroundings and came on strong in the last 10 minutes of the first half.
Ayoze Perez had their first real chance in the 35th minute when he brought a save out of Tigers keeper Allan McGregor from a Massadio Haidara cross.
Then the visitors seized their chance when Robertson presented Cabella with his golden opportunity and the Frenchman took advantage by firing low past McGregor.
Driving sleet added an extra twist to the proceedings moments after the opener but Hull fans had their hopes mistakenly warned in stoppage time at the end of the first half.
Elmohamady bundled a Ramirez free-kick into the net and wheeled away in triumph only to be booked for his efforts after the assistant referee noticed the Egyptian had quite blatantly punched the ball into the net.
Hull's hopes of getting back in the game early in the second half were once again undone by their own carelessness as Ramirez gave the ball away to Jack Colback in the Hull half.
Colback fed Ameobi who dispensed with any notion of setting up one of his fellow strikers and proceeded to lash a 25-yard effort past McGregor and into the bottom corner of the net.
The second goal at least roused the home side, who laid siege to the visitors' goal and came close when a Ramirez free-kick was headed clear by Ameobi.
Minutes later, Coloccini survived a handball claim in the box but then inexplicably sliced the ball over the head of his own keeper Tim Krul as he tried to hack clear, and was thankful team-mate Vurnon Anita was alert enough to head the ball off the line.
Hernandez flopped a weak header wide and Nikica Jelavic contrived to miss an even simpler opportunity from no more than three yards out in the 64th minute.
With Newcastle in cruise control a third goal was no surprise and it duly arrived in the 77th minute when Gouffran ran unchallenged at the home defence and unleashed a goal-bound shot which was deflected into his own net by the hapless Elmohamady.
Cisse - unsurprisingly sporting a pair of heavy-looking black gloves - came on for the final two minutes but the game was already well and truly won.
Source : PA