Sunderland got the better of last season’s English champions Chelsea 3-2 at the Stadium of Light, while Newcastle played out a goalless draw with already relegated Aston Villa away from home at Villa Park.
If Sam Allardyce’s side win either of their final two matches, then they will remain in the English top flight next season.
Former Sunderland star Kilbane was impressed with Sunderland’s performance against Chelsea and criticised Newcastle’s display against the Villans.
The former Everton man also gave his thoughts on Norwich City’s performance in their narrow 1-0 defeat to Manchester United at Carrow Road in the Premier League on Saturday afternoon.
"Everything Sunderland had - desire and teamwork - Newcastle just didn't have,” Kilbane said on BBC 5 live Sport on Saturday evening (17:24). “
If ever there was a team to be taken down it was Aston Villa. But the Magpies showed very little, and neither did Norwich.
“Manchester United have come away from Carrow Road having been gifted the three points. It's all in Sunderland's hands now.”
Sunderland are now favourites to escape the drop, meaning that Newcastle and Norwich are likely to join Villa in getting relegated from the Premier League.
The Black Cats will take on Everton at the Stadium of Light on Wednesday evening before facing Watford away from home at Vicarage Road next weekend.