The draw maintains Mexico's six-game unbeaten run in the final Hexagonal group stage but the point was more welcomed by the visitors, who had lost five of their previous seven qualifiers at the Aztec stadium.
"Any time you can get a point here that's great," Bradley said. "The mentality and the commitment from every guy was really, really good and now we keep moving ourselves forward."
The result means Mexico remain top of the six-team group with 14 points from six games, while the United States are third with eight points from the same number of matches.
The top three teams qualify automatically for next year's World Cup in Russia and the fourth-placed side go into a play off with a team from the Asian confederation.
U.S. coach Bruce Arena made seven changes from the side that beat Trinidad and Tobago 2-0 in Colorado on Thursday but it was his regular captain Bradley who put them ahead after just six minutes.
Bradley intercepted a loose pass on the halfway line and after seeing Guillermo Ochoa off his line chipped the goalkeeper from fully 40 yards out to give the visitors a sensational lead.
It took Mexico 17 minutes to equalise, Vela picking up the ball on the right wing in a fast counter attack and drifting across the 18-yard box before rifling home past Brad Guzan from 22 yards out.
Mexican midfielder Hector Herrera smacked a free kick off the bar from 30 yards after 70 minutes before Bradley hammered a volley off the post from a similar distance four minutes later but neither side could find the winner.
(Reporting by Andrew Downie; Editing by Peter Rutherford)
