Smith had 25 points and nine rebounds, and rookie Al Horford and Joe Johnson each added 20 points for the Hawks. Horford set a career high.
The Hawks had no defensive answer for Zach Randolph, who had 29 points and 17 rebounds. Jamal Crawford added 21 points for the Knicks.
The Hawks, who committed only four turnovers in the first three periods, had six turnovers in the final period as the Knicks briefly took a late lead.
The Knicks went ahead 91-90 on an inside spin move by Randolph with 2:52 left. The lead held until Bibby's pass set up Smith's go-ahead jam.
Following a missed 3-pointer by Crawford, Bibby's jumper with 35 seconds left gave Atlanta a 94-91 lead. Bibby added two free throws with 11.5 seconds left.
The Hawks, who beat Sacramento on Wednesday in Bibby's first home game, have won back-to-back games for the first time since Feb. 2-6.
Smith committed the Hawks' only two turnovers in the half, but New York led 50-48 at the break as Randolph had 15 points and 12 rebounds.
The Hawks took their biggest lead at 77-69 with 3:26 left in the third quarter before settling for an 81-78 advantage entering the final period.
The Knicks scored the last two baskets of the third quarter and opened the fourth with back-to-back field goals by Fred Jones. Crawford added two free throws to complete a 10-0 run for an 84-81 lead.
The Knicks led 88-83 with 8:01 left before the Hawks answered five straight points, including a 3-pointer by Smith.
