Steven Naismith gave the visitors a first-half lead with a finish from close-range after Nikica Jelavić and Marouane Fellaini caused panic in the Reading area.
However, the Royals levelled through Adam Le Fondre before the 25-year-old scored from the spot to seal a vital three points to lift Reading out of the bottom three.
“We showed too much respect towards Everton in the first half and I just told the players at half-time to raise the tempo of our game,” McDermott told BBC Sport.
“I’ve not been listening to the doubters or reading about the doubters in the newspapers. But I will read the papers tomorrow because we have won.”
Reading striker Jason Roberts also expressed his delight at the Royals’ first Premier League victory of the season after a frustrating start to life in the top flight.
The striker posted on Twitter: “That has been coming for weeks… We never got too low amongst the group… We won’t get too high now… Football is SO MUCH better when you WIN!!!”
ENHANCED ODDS: Get 8/1 on Chelsea, Liverpool and City to win