A Citizens side far from their free-flowing best, showing six changes from their last outing, left it until the 88th minute to find a way through a compact visiting defence.
The form book suggested a clear home win, with City unbeaten in their last 24 matches in all competitions - winning all four in Europe - and the visitors winning just three of their last 14.
That Silva chance aside, City had to rely on a Sergio Aguero shot from the edge of the box and a wayward header from the same player - who recently became the club's all-time leading scorer - in a quiet first half.
Jones was also needed to stop a tame effort from eventual matchwinner Sterling, which he made a mess of by flapping at the ball, while up the other end Sam Larsson curled Feyenoord's best opening wide of the target.
The Rotterdam club, who found themselves three down in the opening 25 minutes of the reverse fixture, looked solid at the back and did venture forward on occasion, with Tonny Vilhena rightly denied a penalty after Ederson won the ball but followed through.
City, looking to get the job done ahead of the December 6 trip to Shakhtar Donetsk - a few days before the Manchester derby - came close early in the second half through Kevin De Bruyne and Aguero drives.
De Bruyne then dragged a shot past Jones and past the post, before being withdrawn with a yellow to his name that rules him out of the final group-stage match.
The Premier League leaders, kept out by the opposition just once in all competitions this season, then saw Yaya Toure send a free kick marginally wide.
Feyenoord could sense a big upset as the final quarter of the match approached, sending on leading marksman Nicolai Jorgensen - too ill to start - and themselves coming close through a well-saved Steven Berghuis attempt.
Aguero brought the ball down well, only to then blast wide, and that looked like being City's best chance to claim all three points.
A stalemate would have been enough to send them through in top spot thanks to Napoli's 3-0 win over Shakhtar Donetsk elsewhere this evening, but Sterling's late strike from inside the box - after an impressive piece of interplay with Ilkay Gundogan - made certain of a favourable last-16 tie and kept City's winning run going.