Oviedo, 26, has agreed a contract until 2020, while Gibson, 29, joins on an 18-month deal - both for undisclosed fees.
The pair have been fringe players under Toffees boss Ronald Koeman this season.
Sunderland's David Moyes has now brought in three players this month - including defender Joleon Lescott - who he also signed when he managed Everton.
Midfielder Steven Pienaar and forward Victor Anichebe, who also played under Moyes at Goodison Park, have already moved to Wearside this season.
Another former Everton player, Jack Rodwell, who Moyes sold to Manchester City before he was transferred to Sunderland, was already at The Stadium Of Light.
The Black Cats - who are bottom of the Premier League, three points adrift of safety - host third-placed Tottenham on Tuesday (19:45 GMT kick-off).
Gibson has made just one appearance for Everton this season - as a substitute against Yeovil in the EFL Cup - while Oviedo has played seven times.
Koeman said he was keen to keep hold of Oviedo, but has allowed him to leave after talks with the player.
He said: "With Oviedo it was difficult. He is the back-up for Leighton Baines. He is a perfect professional. He wants a new challenge to play at Sunderland.
"I had told him no way, no chance - but I changed my mind.
"If we get a problem at that position we have some good young ones."
Moyes said Oviedo "is a very good player and he'll help us get forward - he's quick and overall he's a decent left-back", while describing Gibson as a player who would "add a bit of quality with his passing".
Keep up to date with all the Premier League transfers in January as we track and profile all the players leaving or joining a club this month.
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Sunderland have signed Republic of Ireland midfielder Darron Gibson and Costa Rica left-back Bryan Oviedo from Premier League rivals Everton.