Jeremey Senglin, who scored Weber State's last 11 points, made a 3-pointer with six seconds left. Byrd was fouled as he went up for a layup and made the first of two free throws.
Thomas Neff's 3-pointer with 6:48 left gave the Pioneers (7-3) the lead for good at 58-55.
Denver looked to have the game in hand after Joe Rosga made two free throws with 28 seconds left for a 66-62 lead, but he fouled Senglin on a long-range shot and Senglin made three free throws. Denver's Jake Pemberton sank two free throws before Senglin made his final 3-pointer.
Nate Engesser matched a career-high 26 points, making six 3-pointers. Denver hit 13 3-pointers, the 295th straight game with at least one.
Senglin and Joe Bolomboy had 14 points each for the Wildcats (5-5).