Saturday's 0-0 stalemate with Swansea City was Sunderland's fifth draw of the campaign, leaving them with five points from six top-flight outings.
However, Jones - who moved to the Stadium of Light from West Brom in the close-season, claims the Sunderland dressing room remains upbeat.
"I don't think there's any worry," he told the Sunderland Echo. "We know we're very close to getting that [their first win].
"We've been consistent with our draws, and hopefully we can do the same with wins; get one and go from there.
"The effort was there. At times we maybe lacked that bit of quality with the final pass [against Swansea].
"But we worked hard and we got more of the ball in the second half and with that, chances come.
"We created the most clear-cut chances in the second half. On another day, we'd be talking about a great 1-0 win."
