Diamonds shooter Erin Bell slotted two goals for eight points and Caitlin Thwaites shot one from the pink zone for four points during the quarter to lift Australia to a 32-29 lead.
Needing three points for a draw, Tutaia twice had attempts at goal to steal the match but her luck from more than five metres ran out in the dying minutes.
"Dad always says, don't waste time passing to the other shooter, so why not [take the shot]," Tutaia said.
Diamonds captain Susan Pettitt said she was happy her side recovered from a slow start to progress into day two undefeated.
"It was a good game and it was a close one with Maria shooting three six pointers -- that hurt -- but I think the girls played really well.
"Our shooters were shooting really well as well, so coming into our power play, we could nail some four pointers.
"This game is so unpredictable, you just don't know what to expect."
Earlier on day one, defending champions New Zealand survived to hang on 36-35 in a tight game against England before bouncing back to thump South Africa 49-20.
Australia recovered from an opening match scare against Malawi to win 24-23, before toppling Jamaica 35-19, thanks to a 16-point haul from Thwaites. And South Africa sank a three-pointer on the buzzer to draw 29-29 with England after earlier being upset by Jamaica 35-32.
Today's action sees the Silver Ferns take on Jamaica and then Malawi in pool play before the finals, where fifth plays sixth, third takes on fourth, and first plays second for the title. The Diamonds are vying to become holders of all four international tournaments and have never won the Fast5 World Series.
