The Western Cape side secured a 22-15 victory against the visitors, picking up only their third win in seven games this season.
"It shows the quality and character in our side," Coetzee said.
VIDEO: Stormers v Sharks, highlights
GALLERY - Super Rugby - Week 9
"It shows that the way we play and what we want to do is the right way as long as you pitch up with the right attitude and physicality.
"This team has been under tremendous pressure. There are youngsters in the team and we have a long injury list, so they find themselves in a position where they are playing at Newlands in front of 40 000 people."
While the Stormers were lying 11th in the overall standings, Coetzee was delighted to pick up valuable points as they closed in on a host of mid-table sides.
"This was important because we had our backs against the wall, to be honest," he said.
"The way we played was the only way out, to fight back, and we did that well this evening."
The Stormers have a bye in the next round before they play four matches in Australasia, and Coetzee said they would set off with loads of confidence in their ranks.
"A lot of teams must still play tough opposition and the competition is not even halfway yet.
"Our back-up players like a Martin Bezuidenhout, Michael Rhodes, Gerbrand Grobeler, Gary van Aswegen and even Kurt Coleman got a run tonight. It was pleasing to see that they all stood their ground and did well.
"They were all part of this victory and it showed that the depth is good.
"When the top players come back they can change things around and it's pleasing to know that Bryan Habana, Rynhardt Elstadt and Gerhard van den Heever will tour with us, and maybe even Elton Jantjies."
Sharks coach John Plumtree said his team failed to rise to the occasion.
"We struggled and we did not create," Plumtree said.
"There were opportunities (to score) but the Stormers scrambled well.
"We tried to change the way we attacked but we didn't adjust quickly enough to different tactics, and at times we ran out of patience.
"We lacked a big ball-carrier, and this was the type of onslaught we expected, but we are a bit irritated that we did not show up more.
"The Stormers responded well to the pressure but we've learnt a lot from this loss."