Chicago (8-9-16) still must win in its last game, and have New York lose, plus make up a five-goal differential in the process to catch the Red Bulls for the 10th and final spot. D.C. United (9-12-11) and Portland also remain in contention.
Grazzini sparked the rally, taking a layoff feed from Gonzalo Segares and finishing past a diving Bill Hamid from about 15 yards.
Segares then set up the game winner, beating the offsides trap to reach a ball on the left and cross to Chaves, who finished past a stranded Hamid.
United inducted former midfielder Richie Williams, who won MLS Cups with the team in 1996, 1997 and 1999, into its Hall of Tradition during an on-field halftime ceremony.