Adam Le Fondre scored twice as McDermott's side came from behind to secure a 2-1 victory over high-flying Everton.
Steven Naismith had put Everton ahead but Le Fondre levelled with a second-half header before scoring the winner from the spot after being brought down by Seamus Coleman.
McDermott said: "Everyone was saying 'You haven't won, you haven't won' and that can play on people's minds and I obviously know that.
"But all you can do is concentrate on playing the game, enjoying the games and then the results will come because I know with the group we have got there was never any fear that they couldn't do it.
"Obviously it is a relief, I'm just so pleased for the players tonight [Saturday], that has been coming for quite some time.
"Certainly as the manager it (the winless streak) is not relevant, all that is relevant is the points you have got on the board."
The win lifted Reading out of the bottom three, although McDermott felt his side were not at their best until the second half.
"First half we were a little bit too respectful, in the second half I thought we deserved to win," he said.
"In any sport what you need to do is stay in the game, I just said (at half-time) that you are right in the game at 1-0 and if we can raise our tempo and play the way we can it will give us an opportunity.
"I thought we saw the game out really well, we have been working on that as well. We have been in the lead on a number of games in this league but haven't seen them out."
Everton manager David Moyes refused to blame referee Martin Atkinson after seeing his side throw away the lead.
Although he had no complaints over the award of the penalty, Moyes felt his side should have been given at least one themselves.
"I'm not angry with the referee, he should have given a penalty in the first half...but he doesn't give it," Moyes said.
"But I'm more angry with my team because we should have scored and it is up to us to do it.
"We didn't punish them and left the door slightly ajar for them to come in and they did do.
"I think certainly we should have had one penalty kick, I have got no problem with the Seamus Coleman one, it is a definite penalty for Reading."
Le Fondre's equaliser came courtesy of a free-kick from Nicky Shorey and Moyes pinpointed the conceding of set-pieces as a factor in Reading's fightback.
"We gave away a couple of corner kicks and a free-kick in the opening 10 minutes of the second half, we encouraged pressure and gave the ball away a couple of times in our own half," he said.
"Quite simply we didn't pass the way we did in the first half and we gave them encouragement. I have got to give Reading credit, they upped it and tried to do better, which they did do."
Moyes felt his side created enough chances throughout the game, adding: "We are not scoring the goals; when we get the opportunities to put the games away we are not taking them.
"The result should have been the other way but it wasn't and I have to give Brian and his team credit, they hung on and kept going and that is what you have to do in the Premier League, you have to get wins maybe when you are not always expecting to and Reading did that."