Kelly (4-3) scattered four hits and three walks over six scoreless innings, striking out six while placing 67-of-105 pitches for strikes.
Seth Maness, Randy Choate and Michael Wacha each hurled a shutout relief inning to complete the whitewash as the Cards bounced back from a 7-0 loss in Friday's three-game series opener.
Second-place St. Louis came into the contest trailing National League Central-leading Pittsburgh by three games.
The Cubs' Travis Wood (7-10) was tagged with the loss, allowing four runs on four hits over 5 1/3 innings.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys