The 27-year-old right-back turned down West Brom's offer of a new three-year deal on improved terms in favour of a move to the Stadium of Light.
Jones was a fans' favourite at The Hawthorns but had to fight for a first-team spot with Steven Reid and made 22 appearances last season.
The former Preston defender believes his best years are ahead of him and he told the Express & Star: “It’s time to kick on now.
"I know I can play at this level and play consistently well at this level and I feel the next four years should be the best football I play in this league. That’s what I expect and that’s what the gaffer (Gus Poyet) expects as well.
“As a defender I’m proud to keep clean sheets but I feel that I’ve improved offensively and from speaking to the gaffer and the way he wants to play the game it should be a good match.
“I just hope to first and foremost be solid but from a personal point of view I hope to get some assists as well.”
Jones hopes Sunderland's resurgence in form at the end of the season, which saw them win four of their last five Premier League games to dodge relegation, is a "sign of things to come".
He added: “Sunderland got safe when many wrote them off. I know at one point at West Brom we looked at Sunderland and thought ‘yeah, they might be down’ but what they did to turn it around was unbelievable.
“The consistency shown at the back end of last season was a sign of things to come.”