Alan Pardew's side benefited from a hopeless piece of defending to win the second leg 1-0 for a 1-0 aggregate win after last week's 0-0 draw in the first leg in Newcastle.
Home midfielder Papa Gueye was guilty of somehow overhitting a back-pass to goalkeeper Oleksandr Goryainov, who needlessly brought down Moussa Sissoko with the visiting midfielder running away from goal.
That moment of panic saw Goryainov booked and Ameobi punished both him and Metalist by planting the ball to the goalkeeper's right on 64 minutes as Goryainov headed left. It was his 15th goal in Europe for Newcastle.
Newcastle needed several exceptional saves by Krul in the closing 20 minutes, but their reward is a match with Anzhi Makhachkala in the last 16.
It was no more than the Magpies, who had two goals disallowed in last week's first leg on Tyneside, deserved from a spirited display, although goalkeeper Krul, who had denied substitute Willian six minutes later, also had to pull off a stunning double save to keep out Jonathan Cristaldo 13 minutes from time.
Anzhi drew 1-1 with Hannover on the night, with the Russian side going through 4-2 on aggregate after their 3-1 first-leg victory in Moscow.
Newcastle arrived in Kharkiv knowing they needed to avoid defeat to at least take the tie into extra-time, and that a score draw would send them through.
Pardew made five changes to the side which could only manage a 0-0 draw at St James', and that meant a debut for January signing Massadio Haidara at left-back and starts for Danny Simpson, Vurnon Anita, Sylvain Marveaux and Ameobi.
It was the visitors who made the brighter start as the physicality of Ameobi and Sissoko, as well as the control exerted by Yohan Cabaye and Anita in the middle of the field, caused all kinds of problems for the home side.
Ameobi saw a second-minute cross taken off Papiss Cisse's toe as he prepared to shoot after the former had left defender Marco Torsiglieri for dead wide on the left, and Sissoko might have opened the scoring with quarter of an hour played.
He controlled Cabaye's long ball out of defence superbly before cutting inside Torsiglieri, but his weak left-foot shot was easily fielded by Goryainov.
Ameobi's 23rd-minute flick-on caused panic in the Metalist defence once again, but the ball would not sit down for Cisse and it was eventually smuggled away.
The Magpies were enjoying much the better of the game as the temperature plummeted, and Goryainov had to punch away a Cabaye free-kick as Ameobi prepared to meet it at the far post three minutes later.
But the Ukrainian side responded and went close twice within nine minutes through Cristaldo.
The striker headed inches wide from Cristian Villagra's 29th-minute cross, and then saw a 38th-minute effort from Jose Ernesto Sosa's corner blocked by defender Mapou Yanga-Mbiwa.
Newcastle could have gone in at the break in front after Ameobi turned Sissoko's pass into the path of Cisse. The Senegal international had a clear run on goal, but just as he did on several occasions on Tyneside, 37-year-old Goryainov proved equal to the task to keep him at bay.
Both sides returned knowing a tense 45 minutes lay ahead, and it was Metalist who made the brighter start as they attempted to pressurise new boy Haidara.
Newcastle were careless in possession as the half got under way with Sissoko and Cisse gifting the ball to the hosts to stall any momentum, although midfielder Cleiton Xavier blazed a 53rd-minute free-kick high over, just as he had done during the first half, to let them off the hook.
However, Marveaux was just as profligate three minutes later when he exchanged passes with Sissoko, but blasted high over from a tight angle with a square ball perhaps the better option.
But the Magpies made the breakthrough with 26 minutes remaining when Sissoko's industry was finally rewarded.
The midfielder chased down a poor back-pass by defender Papa Gueye and in his efforts to rescue the situation, Goryainov tripped the Newcastle man.
Belgian referee Serge Gumienny pointed straight to the spot and after Sissoko had received treatment, Ameobi kept his nerve to send the goalkeeper the wrong way.
His strike was greeted by a deafening silence as the 500 travelling fans celebrated knowing their side had one foot in the next round.
But if they needed a reminder that there was work still to be done, Krul had to come from his line to deny Willian with 20 minutes remaining as Myron Markevich's men launched a late assault.
The Dutchman excelled once again seven minutes later to prevent Cristaldo from levelling twice within seconds as Pardew's men held firm to secure the victory they wanted.
MAN OF THE MATCH
Tim Krul - Newcastle: Superb handling skills and agility ensured there would be no recovery from Metalist.