The Lady Raiders (14-0, 2-0 Big 12) are off to their best start since the 2003-04 season, when the team started 15-0.
Saturday's win was Texas Tech's first in Columbia since 2004, with both teams trading home wins the past seven seasons.
Neither team could separate itself throughout the first half and into the second, but Brown's two free throws with 4:19 remaining capped an 11-0 run by the Lady Raiders, giving them a 62-53 lead.
Missouri's Sydney Crafton converted a three-point play with 52.8 seconds left to narrow the deficit to 66-63, and after a missed Texas Tech 3-point attempt, the Tigers had a chance to tie the game with 15 seconds remaining.
But Missouri's Christine Flores missed from long range and BreAnna Brock failed to put in a layup after grabbing the rebound.
Casey Morris then iced the game with two free throws with 12.4 seconds remaining. Flores led Missouri (10-3, 0-2) with 16 points.