Peter Crouch's 100th Premier League goal had handed the home team the lead inside just seven minutes.
But Everton equalised when Seamus Coleman's shot deflected off Stoke captain Ryan Shawcross and after some confusion and an initial offside decision, Craig Pawson awarded in the Toffees' favour.
Koeman's charges had opportunities to win the game in the second-half, substitute Ademola Lookman testing the reflexes of Lee Grant while the 19-year-old flashed across the face of goal after Tom Davies' header was saved.
Yet despite drawing, the Everton boss believes his side can take heart from their performance.
Blues boss "pleased" with his side's display
He said in his post-match press conference: "Yeah, maybe on the end yes [we could have won] but overall, I think it's a fair result.
"For the first 45 minutes, maybe we had more and better ball position but really the clear chances were for Stoke. Good saves by Joel [Robles].
"Second-half was better, but it's always difficult to have good defence against Crouch because it's long balls and then the second balls. But I think overall we did defensively well.
"Maybe the second-half we had to be more a little bit more clinical in our ball position. We lost stupid balls in the midfield. But overall, I'm pleased with performance of the team.
"I think we played good football and everybody knows it's always a difficult place to come and we take one point. If you can't win, don't lose, and that's good."
On his decision to change to a back-four at half-time, withdrawing teenage defender Mason Holgate as James McCarthy was introduced, Koeman said: "I think we had space to play but they had maybe two, three big chances in the first-half even against five defenders.
"That wasn't good. I saw really reasons enough to change system, to put James in midfield and I think we control a bit better the midfield [after that]. That was the decision to change at half-time."
The Dutchman also sympathised with those who felt frustrated with the manner of Everton's goal, which was awarded a couple of minutes after the ball first hit the back of the net.
"It's difficult," he said when asked for his thoughts. "I can understand that the people of Stoke, even Mark [Hughes], were disappointed because it was strange.
"First everyone expected it was offside. Finally, when I was watching the clip, then it's easy. It's really a goal. Good, right decision from referee, but he took his time and that was difficult to understand."
Koeman says Lookman must focus on being further forward as he hails Schneiderlin
January signing Morgan Schneiderlin made his first start for the club in midfield, while fellow winter addition Lookman once again starred from the bench, and Koeman was pleased with the pair's displays.
On Schneiderlin, he said his performance was "good" and explained: "Of course I know him, that's the performance that he can give the team.
"Really strong, comfortable on the ball and even he can play in a different position than he did today, more offensively in midfield.
"We know we have big competition in midfield. The best will start for that game. It can be different for this Saturday [against Bournemouth]."
And on lively substitute Lookman, who signed from Charlton Athletic for a deal worth around £11 million and again impressed, Koeman hinted that a start could be on the horizon.
"I'm pleased [with his performance] but also today I think he dropped too much back in the 20-25 minutes that he played," he said.
"I told the boy after the game, we need him more between the lines in front of the midfield players. Several times he came back on the same line as our midfield players.
"If he can bring that speed, that quality, more up on the pitch then he's more dangerous. The boy is a young boy and he needs to learn.
"With game-time, he will improve. Okay, it's possible in that next couple of weeks that he can start because he's that quality of player."