The Royals looked destined for a 1-0 loss at the Madejski Stadium on Saturday afternoon before Adam Le Fondre's last-minute penalty rescued a point.
With trips to Chelsea and Sunderland next up before the international break, McDermott was delighted to avoid the prospect of going into the next home game in a month's time yet to get off the mark.
"It's a strange thing to say but, psychologically, it's important that we got that result today," he said.
"We know we can compete. We've done that today - there's no doubt about that.
"So it's important to get a result and we have got a result today."
McDermott insisted a draw was "the least" his side deserved. "We'd have been disappointed to have lost the game over the 90 minutes.
"Those of you who have seen Reading play over the last three years know that they keep going.
"We have scored a few late goals in our time."
In truth, Reading showed little sign of bouncing back from conceding an awful opener until Gareth McCleary entered the fray, the new signing changing the game and winning the penalty.
McDermott said: "We had this conversation yesterday about the group.
"It's always going to be about 20-odd players.
"We have two players for each position and, over 38 games, we're going to need every one and I'm convinced every single one of them can play in this league."
That includes goalkeeper Adam Federici, who gifted Stoke the lead when he embarrassingly spilt debutant Michael Kightly's deflected shot into his own net.
McDermott said: "We all make errors. I make errors most days.
"He's a great character, he works really, really hard each and every day. He's a top professional and he's saved us many, many times, and he will do in the future.
"He was upset but that's part of life. You have to overcome your disappointments quickly and he'll do that."
After equalising, Championship winners Reading almost completed a comeback to rival that in their first ever Premier League game six years ago when they recovered from 2-0 down to beat Middlesbrough.
McDermott, who was chief scout back then, said: "I was in Switzerland at the time and I watched the game.
"I got a text to say we were were 2-0 down and the next text said that we were 3-2 up.
"So, it did enter my head."
He added of his first match as a Premier League boss: "I'm not sure 'enjoy' is the word that you'd use being a manager.
"There is a word but I'm not sure what it is."
He added: "I'm pleased it's out the way in a way.
"There's always a lot of hype for the first game of the season but we can all move on now."
Stoke boss Tony Pulis questioned referee Kevin Friend's penalty decision.
Pulis felt the verdict itself - which also saw midfielder Dean Whitehead shown a second yellow card - could have gone "either way". But he was more angry about Friend's decision to play advantage, allowing Reading a shot at goal, before deciding to pull it back.
Pulis said: "I'll speak to [referees' chief] Mike Riley on Monday and find out how long they're going to give.
"They either play on or they call it back.
"It gives the advantage to the attacking team, so it's a good rule.
"It's just being consistent.
"Where does it start? Where does it end?"
Pulis was happy with new signing Kightly, who moved from rivals Wolves this summer, saying: "It's lovely for him, and as an ex-Wolves player as well - important."
But he reserved his biggest praise for defender Robert Huth, who started today despite spending most of last week in hospital with viral meningitis.
"I don't know what they feed them in Germany," Pulis said.
"We're very, very pleased to have him back and he's got a heart the size of a lion."