Andre Gray had earlier given the Hornets the lead in the 13th minute with his first goal for the club before substitute Tammy Abraham equalised for the Swans 10 minutes into the second half.
Much was expected of the pair following comparatively large outlays over the summer, but as both Ronald Koeman and Eddie Howe have found to their detriment so far, such upheaval doesn't always yield instant success- nor is it a guarantee of improved fortunes in the mid-term.
Bournemouth have had problems scoring goals this season.
Wednesday's cup tie was the first of four successive home games so to score three goals and keep a clean sheet was a welcome confidence boost for the whole squad.
Though his place in the Everton side was no longer guaranteed, Barry was a very influential figure in the Toffees' dressing room, which is now smarting after a poor run of results in the Premier League and Europe.
King finished last season as one of the Premier League's form strikers.
Forward Callum Wilson and goalkeeper Adam Federici are still out and will play no part today.
As for Jagielka, he also missed the Bournemouth match with a hamstring injury.More news: The Next Big Focus In The Russia Investigations: Social Media
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Niasse - Roberto Martinez's last signing as Everton boss - was not finished there, however, bundling the ball over from close range five minutes later to relieve the pressure on Koeman's under-performing side.
Everton have faced Bournemouth four times at home in all competitions and come victorious on all four occasions.
Sunderland, meanwhile, name three former Everton men - Jack Rodwell, Darron Gibson and James Vaughan in their starting XI, with another - Bryan Oviedo - on the bench.
Koeman has left out Leighton Baines, Wayne Rooney, Jordan Pickford and Morgan Schneiderlin while Kevin Mirallas is not named in the matchday 18.
Everton nerves were soothed, however, when Lukaku scored on 83 and 84 minutes. This fixture last season proved to be a rollicking goal-strewn affair with the Blues prevailing 6-3.
Calvert-Lewin should have put the result beyond all doubt late on, but his miss ultimately mattered little as Everton held firm to claim a vital win and move out of the relegation zone, where Bounemouth remain second from bottom. With a home match against Burnley and a trip to newly promoted Brighton to follow, three wins has to be the target from these fixtures before the global break interrupts the campaign once again.
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