Normore, a five-time All-Canadian, helped St. Francis overcome a 3-0 second period deficit to advance to the tournament semifinal. The X-Women will face top-ranked McGill in semifinal action Saturday.
"I noticed I was on my own, I went down and had a quick look," Normore said of her winning goal. "I had heard previously to look blocker side but I ended up going glove side, shot the puck and hoped for the best."
Normore finished with two goals, the other coming on the power play to tie the score 3-3. Sarah Bujold and Kara Power had the others.
Kailtyn Mora, Jessica Pinkerton and Jessica Pellegrino scored to stake Guelph to its early lead.
In Thursday's other quarter-final, Melodie Daoust had three goals, two of which came on the power play in the first period, as McGill downed Calgary 8-2.
Gabrielle Davison had a goal and two assists and Leslie Oles, Elizabeth Hillier, Joanne Cagianos and Kelsie Moffatt also scored for the Martlets.
"It was a great feeling to come out strong and get a win in our first game," said Daoust. "It helps settle down the nerves and now we can just focus on playing our game. We did a pretty good job on the power play today and created lots of scoring opportunities. I credit my teammates for my hat trick, they made some nice passes and I just had to take the shot."
Iya Gavrilova and Chelsea Peterson replied for the Dinos.
Calgary will face Guelph Friday in consolation action.