Kolb (29-of-48 passing, 324 yards, two interceptions) hooked up with Andre Roberts on a score-tying, 15-yard TD toss with 22 seconds left in the fourth quarter, sending the contest into extra time.
Feely's subsequent 46-yard game-winner boosted the Cardinals to their first 4-0 start since 1974, when the team (then located in St. Louis) began the year 7-0.
Roberts finished with six grabs for 118 yards and two scores. Larry Fitzgerald chipped in with eight receptions for 64 yards and a touchdown.
The Dolphins (1-3) got a team rookie-record 431 passing yards from Ryan Tannehill, but it wasn't enough as they fell in overtime for the second straight week.
Cameron Wake was credited with 4 1/2 of Miami's club-record eight sacks.