The Black Cats came within seven minutes of inflicting a hugely damaging defeat on the Magpies at St. James' Park in Sunday's Tyne-Wear derby before Aleksandar Mitrovic's late header rescued a point for the hosts.
That draw leaves both North East clubs in the relegation zone, with Newcastle a point and place behind Sunderland but with both clubs holding a valuable game in hand over 17th-place Norwich City.
Jenas, who played 152 times for the Magpies between 2002 and 2005, believes that the appointment of Rafael Benitez has given his old club a better chance of survival but he still believes it is Sunderland who are better equipped to stay up.
The 33-year-old wrote in his Yahoo Sport column: "A lot is made of Rafa's ability to get teams organised and, while I haven't seen huge evidence of that yet, what he also brings is a calm head on the touchline. It would have been easy to panic going 1-0 down in the derby.... but Benitez was incredibly patient. He made the right substitutions at the right times and it paid off in the end.
"Overall I thought it was a good derby. It may not have been top quality football, but in terms of what I've seen from Newcastle this season, it was chalk and cheese. I saw hope and effort and work-rate, but the problem is that those things only get you so far. Newcastle have lacked quality all season, and not even Benitez can change that at this stage.
Meanwhile, Sunderland look organised, they have strength in depth and they've got a striker [Jermain Defoe] who will put the ball in the back of the net when they need him to. Plus, I think it's only a matter of time before they start getting clean sheets. As a former Newcastle player it hurts me to say it, but I think if anyone is going to stay up out of those two teams, it will be Sunderland."
Sunderland would be in an even better position to avoid relegation had they not conceded a late equaliser for the second match in a row, after Virgil van Dijk scored in the 93rd minute to deny them a 1-0 win at Southampton last time out.
For Newcastle, meanwhile, it is just one point from their first two matches under Benitez, and though that is better than none, the Spaniard needs to start leading his new side to wins sooner rather than later if they are to have any hope of retaining their Premier League status.