The Renegades had posted 222-4 in Australia's Twenty20 competition but their rivals chased it down with 223-8.
Ben McDermott, 22, set up the victory with 114 from 52 balls for Hobart.
Needing 16 off the last over to win, Broad hit 11 from four balls - including two fours before a single off the last ball in a dramatic finish.
Victory boosts the Hurricanes' chances of reaching the semi-finals. They move up to fifth in the standings, with the top four progressing.
England's Stuart Broad hit the winning runs for Hobart Hurricanes as they beat Melbourne Renegades with the highest score in Big Bash history.