The draw gave D.C. second place in the Eastern Conference.
United (17-10-7) went 5-0-2 in its last seven matches to finish with 58 points, one more than New York and Chicago (17-11-6). New York finished third, ahead of fourth-place Chicago, on a tiebreaker.
With New York beating Philadelphia earlier in the day, the Fire needed to beat D.C. to avoid playing in the first-round one-game elimination round.
Pajoy beat Sean Johnson with a diving header in the 50th minute.
Pajoy's eighth goal of the season, and third since joining United from Philadelphia, was one of D.C.'s few good chances. United was held to two shots on goal in the first half and five for the game, including a carom by Pajoy in the 70th minute that Johnson cleared off the line.
Chicago took nine shots on goal, but only forward Patrick Nyarko beat Hamid. He right-footed a 24-yard shot low to Hamid's left side in the 16th minute for his fourth goal of the year.
Every opportunity after that, the Fire either misfired or squandered. In the last 25 minutes, Dominic Oduro, Serjill MacDonald and Jalil Anibama all had close-in opportunities that were either saved by Hamid, hit the post, flew over the crossbar, or, in the case of Oduro's last shot, were blocked by midfielder Nick DeLeon. Hamid's eight saves matched his season high.