Christian Eriksen reduced the arrears in fortunate circumstances, his initial shot blocked by Aymeric Laporte before deflecting back off the Dane and past the stranded Ederson, with Sterling adding the third in the second half.
Two goals from Wilfried Zaha and one from James Tomkins put Palace ahead at the interval but having conceded twice too they were forced to scrap for the three points that took them six clear of the drop zone.
Zaha's second, a bullet header from Luka Milivojevic's cross, was particularly impressive.
Zaha restored the two-goal advantage with a bullet header from Milivojevic's flawless cross and, in a frenetic opening 45 minutes, Jose Izquierdo struck with a fine finish for the visitors.
After a boring 90 minutes against an out-of-form Hornets side, Ince hit the deciding effort to take David Wagner's side seven points clear of third-bottom Southampton.More news: Russian Federation vetoes U.S. bid to form new Syria chemical weapons inquiry
More news: Trump Declares "Mission Accomplished" After Targeting Syria in Friday Night Missile Strike
More news: Diane Abbott: We will only back conflict in WW2 scenario
What we will do is talk about the second half and the fact we gave ourselves a chance but it's nearly impossible to turn it around when you give away the goals we gave away.
Olivier Giroud came off the bench to inspire the comeback, scoring either side of an Eden Hazard equaliser after Dusan Tadic and Jan Bednarek had struck for Southampton.
Hodgson's side are now three points clear of the drop zone, although 18th-placed Southampton do have a game in hand.
Burnley put Leicester City to the sword within the first 10 minutes of the match as the home side cruised to a 2-0 lead against at Turf Moor. "On their day they are a very good side and we'll have to be mindful of them as a team, not just as individuals, but you have to be conscious of Zaha and the form he's in".
Jamie Vardy netted the consolation for the visitors.
The Clarets are only two points behind sixth-placed Arsenal, who face Newcastle at 13:30 BST on Sunday.