William Eklund and Jacob MacDonald scored, and Blackwood made 38 saves for the Sharks (18-51-9), who have lost four of five (1-3-1).
"We just cheated the game," San Jose coach David Quinn said. "I thought in the second and third we were wanting it to be easy. We were hoping that things were going to happen instead of approaching the game the honest way and the right way. I thought we were fortunate to get into overtime."
Eklund put San Jose ahead 1-0 at 7:22 of the first period with a backhand in front off a pass from Fabian Zetterlund on a rush.
MacDonald increased the lead to 2-0 at 2:22 of the second period. He scored with a backhand from the low slot after Mike Hoffman knocked the puck away from Yegor Sharangovich at the blue line.
Andersson got Calgary to within 2-1 at 5:35, taking a pass from Matt Coronato in the left circle before scoring on his own rebound in the slot.
Kadri tied it 2-2 at 14:51, roofing a shot from the right circle on a power play.