This game was gifted to the Patriots by Falcons coaching and QB incompetence regarding time-mismanagement (loss of timeouts), and horrible play-calling (pass plays that resulted in missed opportunities or turnovers).
Timeouts: The Cardinal rule in NFL football is to protect your timeouts in the 2nd half (EVEN WHEN UP 25 pts!!). Had they followed the rule, they'd of had 2 timeouts with 57s to drive 65 yards for a possible FG try. Instead they muffed it, over-hurrying with Ryan making poor pass decisions resulting in OT.
Play-calling: Late in the 3rd up 16pts the Falcons elect to pass when runs and a simple punt put would eat valuable playtime and give back field position for the defense. The backbreaker, as mentioned by another, though came later around 4 min left in the 2nd and 11 up 8 on the NE 22, QB or Coaching decide to pass the ball when easily in FG position resulting in sack/push out of FG range. They'd made the same erroneous decision twice in the game.
This was a coaching meltdown first and foremost, while Bellichek and company took full advantage. Hard to watch.