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."
He added: "Obviously it is a relief, I'm just so pleased for the players tonight, 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."
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."