Cabaye responded at the other end by swerving in a threatening long-range shot that Pantilimon pushed away, before a dipping left-footed shot from substitute Sako went just wide.
Sunderland brought on Jeremain Lens and Duncan Watmore in an attempt to find an attacking spark, while Patrick Bamford came on for Palace after Sako lasted only 23 second-half minutes before having to come off through injury.
Lens had a left-footed effort saved by Hennessey after the lively Watmore had created an opening and Wilfried Zaha shot narrowly wide as the game increasingly looked like it was heading towards a draw.
But after Bolasie gave away possession, one throughball from Billy Jones caused a mix-up between Dann and Hennessey, allowing Defoe to steal in and tap home what proved to be the winner.
Pantilimon did well to save from Cabaye to preserve three vital points for the visitors, whose victory was soured slightly when goalscorer Defoe had to limp off in the closing moments.
Crystal Palace manager Alan Pardew: "We overdid it a little bit in certain areas. We should have been more clean and clinical in the final third and we would have created more chances. They asked us questions we needed to answer if we are going to go forward and evolve into a different type of team. We struggled - it just didn't happen.
"I just didn't think we hurt them enough and we couldn't get in behind. The longer it went on the more belief Sunderland got."
Sunderland manager Sam Allardyce: "It is a massive result for us and it is no coincidence that zero goals against has got us a victory. Our two clean sheets of this season have got us six points, so it is the way to go and really always has been the way to go when you play the game of football.
"We frustrated the crowd and frustrated Crystal Palace and then we come out with a fantastic three points and a great victory for us - our first away win of the season."
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