It was a challenging game for the Winnipeg skip, who stole a deuce in the 10th end for the win.
"Not exactly to plan, we struggled a little bit with the ice but a win's a win and we found a way to win the game," Jones said. "We'll have to be sharper tomorrow."
Jones and her team of third Kaitlyn Lawes, second Jill Officer and lead Dawn McEwen scored two in the first end and led 5-2 at the midway point. Finland came back with three points in the sixth end and sat two with hammer in the 10th after Lawes hogged a guard attempt.
However, Lawes responded with a hit-and-roll behind cover that set up Canada's steal. Puustinen rubbed on a guard on her hit attempt to force an extra end.
"She made a big shot when she had to, and she always does," Jones said of Lawes, who is making her world championship debut. "That was the thing about this game — when someone missed, and we did have a few too many misses out there, the next person would pick up and make a good shot and that's what it's all about."
Canada had never lost to Finland in 12 previous matchups at the world championship. Jones will have a tough challenge on Sunday with games against 2013 world champion Eve Muirhead of Scotland and four-time silver medallist Margaretha Sigfridsson of Sweden.
Jones beat Sigfridsson to win gold at the Sochi Games last year.
"We're playing two amazing teams that we have a ton of respect for," said Jones.
Switzerland's Alina Patz and China's Sijia Liu both posted a pair of victories to top the standings after two draws. Russia and Scotland were tied with Canada in third place at 1-0.
Germany was 1-1 while Denmark, Finland, Norway, Sweden and host Japan were 0-1. The Americans were last in the 12-team field at 0-2.
The top four teams at the end of round-robin play will advance to the Page playoffs. The medal games are set for March 22.
A Canadian team hasn't won gold at the women's worlds since Jones took the 2008 title with Cathy Overton-Clapham at third. Lawes replaced Overton-Clapham in 2010.
Jones, a five-time national champion, made her world championship debut in 2005. She won a world silver in 2010.
Canada leads all countries with 15 gold medals and 31 podium appearances at the world championship since 1979. Sweden is next with eight gold and 23 medals in total.