The Giants opened the scoring when Burns pounced on a rebound at 4:36, but the Hurricanes equalized at 13:55 when Graham Hood set up Russell Maxwell on a 2-on-1.
The Hurricanes took the lead in the second when Daniel Johnston pounced on a loose puck in the slot to beat goalie Liam Liston.
Ten minutes later the Giants tied it on a broken play, as Carter Popoff scored his first WHL goal to make it 2-2 headed to the third.
Axel Blomqvist was stopped by Liston on a breakaway in the third only to get his revenge shortly after and give Lethbridge the lead back at 4:46.
With the goalie pulled for the extra attacker in the final minute, Dalton Sward tipped in a point shot to tie the game with just 50 seconds left.
The win moves Vancouver to 4-10-0 while Lethbridge suffers their first shootout loss and now drops to 8-7-2.
Hurricanes goalie Ty Rimmer finished with 33 saves. Liston stopped 26 of 29.