Carver was confirmed as Newcastle boss until the end of the season in midweek - having taken the reins following Alan Pardew's departure earlier in January - and celebrated a victory at the fifth attempt to keep Hull in the Premier League's relegation zone.
Cabella gave Newcastle the lead five minutes before the break when he capitalised on poor defending to open his account for the club with an impressive strike from distance.
There was more drama to come on the stroke of half-time as Ahmed Elmohamady appeared to have levelled - only to have his goal chalked off by referee Phil Dowd due to a blatant use of his hand after consulting his assistants.
Ameobi doubled the visitors' advantage five minutes into the second half when he let fly from distance to compound frustrations at the KC.
Gouffran's solo run and deflected strike 12 minutes from time ensured the points belonged to Carver and Newcastle, leaving Hull with work to do and Steve Bruce looking to the potential arrivals of Aaron Lennon and Dame N'Doye to brighten moods.
In a lively opening, both sides went close but it was the hosts who edged the early stages with Gaston Ramirez in particular causing problems for Newcastle, Fabricio Coloccini clearing a Curtis Davies header off the line from one of the Uruguayan's deliveries.
Carelessness in possession and fouls from both sides soon became a common theme only for Cabella to give Newcastle the lead with the game's first moment of real quality.
Sloppy play from Andrew Robertson handed the Frenchman possession, with Cabella shimmying left and firing past Allan McGregor.
Elmohamady did his best to level the scores but saw his cheeky effort ruled out, picking up a booking for his troubles.
Loose play in possession again cost Hull shortly after the break as Ameobi made the most of a careless pass to stride forward and take McGregor by surprise - finding the far corner with a stunning strike from 30 yards.
Coloccini almost gave Hull a way back into the game with a bizarre hooked clearance that nearly ended up in his own net but Vurnon Anita was able to head clear.
It sparked a rare period of Hull dominance but - despite Bruce being able to recall Nikica Jelavic and Abel Hernandez from injuries - a lack of creativity going forward proved the hosts' downfall, with both strikers missing from close range before Newcastle killed the game off.
Gouffran exposed Hull's backline once more to burst into a dangerous area and round off the scoring with a somewhat fortuitous strike that looped in off the luckless Elmohamady.
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