The Premier League strugglers announced that Oviedo had signed a contract to 2020, replacing Patrick van Aanholt who completed his switch to relegation rivals Crystal Palace earlier in the day.
Moyes signed the 26-year-old for Everton while boss of the Merseyside club before an ill-fated move to Manchester United in 2013 as Alex Ferguson's short-lived successor.
Gibson, 29, also played under Moyes at Goodison Park and has been reunited with the Scot on an 18-month deal. No financial details were given.
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"Bryan is a very good player and he’ll help us get forward," Moyes told the Sunderland website (www.safc.com).
"I signed him for Everton from Copenhagen as cover for Leighton Baines at the time, and he’s only not getting in because of how well Leighton’s done."
Oviedo has made seven appearances for Everton this season.
The two players joined a growing list of former Evertonians on the Sunderland payroll.
Former England defender Joleon Lescott signed this month while England midfielder Jack Rodwell, South African midfielder Steven Pienaar and Nigerian striker Victor Anichebe are also on the books of the Wearside club.
(Reporting by Alan Baldwin, editing by Ed Osmond)
