In front of a febrile crowd at Melbourne Rectangular Stadium, Raso struck in the ninth minute with a low, right-footed shot and doubled down in the 39th from a goalmouth scramble as unused captain Sam Kerr cheered by the touchline.
Fowler made sure of the win with a deft touch in the 58th minute that pinged in off the right post, before stand-in skipper Steph Catley slotted a penalty in stoppage time as a relieved nation celebrated the co-hosts' advance.
Australia had needed three points from the match to advance after their shock 3-2 defeat by Nigeria.
"It means so much to us," Raso told Channel 7. "I'm so proud of the girls.
"We had our backs against the walls after (the) last game and we knew that we needed to come out and put in a big performance.
"We showed that never-say-die attitude and we gave it all tonight."
Canada, who had needed a draw to advance, bowed out at the group stage for the first time since 2011 and with regrets with the way it unfolded in Melbourne.
Goalkeeper Kailen Sheridan captured Canada's heartache as she broke down in tears before Catley's spot kick.
"Football can be cruel sometimes," Canada coach Bev Priestman told reporters.
"I think the team lacked belief.
"I didn't think the team would be as rattled as we've seen tonight."
For Australia, there was no need for Kerr to be Australia's saviour and the striker can rest her injured calf for a week before the co-hosts' next clash in the last 16, most likely against Denmark.
Australia topped Group B on six points, one ahead of Nigeria, who also advanced after a 0-0 draw against Ireland.
Thrilled Australia coach Tony Gustavsson had been under pressure after the Nigeria loss.
"We played Olympic champions in a do or die game with our skipper, one of the best forwards in the world - in my opinion the best forward in the world - on the bench and we win 4-0. That says a lot about the belief in this team," he said.
Canada's talismanic captain Christine Sinclair started on the pitch but bowed out of her sixth and final World Cup in disappointment at halftime when Priestman triggered a raft of substitutes.
Australia played with courage and were quickly rewarded when a Catley cross found Raso in the box.
Raso drilled a low shot inside the far post to put Australia in front. Although an offside flag went up, the goal was confirmed by VAR, sending the crowd into raptures.
They almost made it 2-0 minutes later, but Raso's shot was well saved by Sheridan.
Home fans were roaring again in the 34th minute as Fowler charged in to thump the ball in from close range, with Raso providing the cross.
VAR again intervened, cancelling the goal for offside.
But five minutes later, Raso was celebrating her second strike after Sheridan failed to clear the ball from a set-piece.
Priestman made changes at halftime, with the retiring Sophie Schmidt coming on and blazing over the bar in the 56th minute.
But another Australian raid minutes later ended with Fowler slotting in a cutback pass from Caitlin Foord to blow out the lead.
Worse was to come for Canada as Jessie Fleming clattered into Katrina Gorry just inside the area to concede a penalty, allowing Catley to blast the ball into the net for the coup de grace.