Marouane Fellaini headed the Toffees ahead with his 12th goal of the season late in an evenly-fought first-half, in which Reading hit the woodwork, and the Merseysiders eased to victory after the break.
Defeat left Reading in the bottom three and in the thick of a battle for survival while Everton, after last week's disappointing loss at Norwich, could still entertain hopes of finishing in the top four.
To their credit, Reading battled on, even if only to reduce the damage to their goal difference column.
Mucha made a good save from Jobi McAnuff and the visitors did pull one back six minutes from time as Harte crossed from the left and an unmarked Robson-Kanu headed home with conviction.
Fans and players were left stunned by Nigel Adkins' departure, with the team having lost only two of their last 12 league matches. Players were told on Friday morning by the captain, Adam Lallana. They now have to put the news behind them and win against Everton at home.
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.
Brian McDermott believes the club's maiden Barclays Premier League victory of the season is a turning point for the club. The win lifted Reading out of the bottom three. More details and all the latest football news can be found on the ITV Sport site here.
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.
"Certainly as the manager it (the winless streak) is not relevant, all that is relevant is the points you have got on the board.
– Brian McDermott, Reading FC Manager - full coverage on the ITV Sport website here