The unlikely duo scored just over two minutes apart in the third period and sent the surging Flames to a 4-2 victory over the Dallas Stars.
Sarich, a defenseman, gave Calgary the lead at 2:35 with his third goal of the season, a snap shot into the top corner from just inside the blue line.
"I don't score them very often so when they really mean something, it makes it that much more satisfying," Sarich said.
Then, with the Flames short handed, Staios jumped up to join Rene Bourque on a rush, took Bourque's pas and fired a long slap shot past Kari Lehtonen for his second goal.
"I saw Bourque going down the wing and I knew that they were tired so I jumped up into the play and I got it and I shot it as hard as I could," said Staios, playing just his fifth game since Dec. 5 due to a combination of injuries and being a healthy scratch.
Dallas coach Marc Crawford was incensed at his team after Staios' goal. He pulled Lehtonen immediately and replaced him with Andrew Raycroft. However, Lehtonen returned to the net after the next stoppage.
"On the short-handed goal, we can't make that play. The game is right there, it's in the balance, we've got to be smart on that," Crawford said. "You're trying to tie it with 15 minutes to go and you don't have to. You just have to keep doing things right. We've got to get that through our thick skulls."
Olli Jokinen and Tom Kostopolous also scored for Calgary, which has used a 10-1-2 run to move into a five-way tie for fourth place in the Western Conference, one point behind the Phoenix Coyotes.
"What is giving us success over the past little while here is we've been taking one game at a time and not getting too caught up in the standings," Staios said. "As you know, it's ninth to fourth in any given night."
The Flames are 6-0-2 in their last eight home games and have four remaining on a five-game homestand. Their next game will be played outdoors Sunday at McMahon Stadium against the Montreal Canadiens in the Heritage Classic.
Mike Ribeiro and Jamie Langenbrunner scored for Dallas, which played without leading scorer Brad Richards for the second game after he was shaken up in a collision on Sunday.
Dallas is 1-6-1 in its last eight and has lost six straight on the road.
"We're going through adversity right now and it's going to make our team stronger. Every team goes through it," Stars defenseman Stephane Robidas. "You look at Calgary early in the year and they were one of the worst teams in the league and look where they're at right now."
By the time Dallas wraps up its road trip Saturday night in Vancouver, the Stars could be out of a playoff spot for the first time in several months.
"It's tough," Robidas said. "I'm not going to lie to you, but we can't focus too much on the standings and what's going on. We have to find our game and get back to the team that we were early in the year.
"Early in the year, we found ways to win games. Now we're finding ways to lose games. That's been the main difference."
Calgary outshot Dallas 16-3 in taking a 1-0 lead in the first period. The Stars played better in the second period and benefited from fortuitous bounces.
The bad luck around the Flames net began when Matt Niskanen's shot, going way wide, caromed sharply off the skate of Calgary defenseman Anton Babchuk and nearly got past goalie Miikka Kiprusoff, who made an alert save.
The Stars tied it two minutes later on another pinball-like goal from Ribeiro.
Ribeiro lost control of the puck on a well-defended 2-on-1, but as he gathered it near the corner and whipped it toward the net, it changed directions off the stick of defenseman Robyn Regehr and bounced in off the shin of Jokinen.
The Stars took the lead at 10:47 on another strange one. James Neal's shot toward the net went off Bourque's stick and zipped straight to Langenbrunner in the slot for his sixth goal.
Calgary tied it at 14:09 when Kostopolous tipped in Rene Bourque's slap shot.
Calgary opened the scoring at 4:14 of the first period when defenseman Mark Giordano's low slap shot from the blue line was deflected in by Jokinen for his 13th goal.
Kiprusoff (26-19-4) made 22 saves in his 12th consecutive start.
Lehtonen stopped 27 shots.
Notes: Dallas D Stephane Robidas played in his 700th NHL game. ... Since the Flames sent Ales Kotalik to Abbotsford of the AHL on Jan. 27, he has no goals and six assists in seven games. ... Calgary LW Curtis Glencross' five-game goal streak ended. ... The Flames are 13-4-5 since Darryl Sutter stepped down as general manager and was replaced by Jay Feaster.