It was Sunderland's third successive win and Keane believes a factor in their form is players returning from injury.
"I've known all along that with a fully-fit squad we'll have a better chance," he said.
"All the players are pushing each other - there's not a better team at training and they are all very competitive and are pushing each other.
David Healy pulled one back but Jones restored the two-goal advantage.
Despite Sunderland fans celebrating in expectation of top-flight football next season, Keane remained cautious.
"We can still be caught," he said. "We are in a much better position than we were three weeks ago but we have tough game (Manchester City) coming up and obviously we are not safe.
"Maybe a few teams last season thought that and West Ham went on an unbelievable run. The likes of Birmingham, Bolton and Wigan are capable of winning games.
"We shouldn't be dancing in the streets after winning three games in a row."
In fact, Keane thought the opener came against the run of play.
"Our performance wasn't great and our passing wasn't up to scratch but we scored at the right time," he said. "In the first half we were very poor, I was glad to get them in at half-time. We huffed and puffed but didn't have quality but we scored just before half-time."