A deflected Darron Gibson strike and a Victor Anichebe goal gave Moyes' men another three points at Goodison Park, their fourth in a row on their home patch.
The Toffees travel to Arsenal on Tuesday, knowing a surprise win will put them right in amongst the scrap for that coveted fourth spot.
"If we can beat Arsenal in midweek, people might start to mention us for a Champions League spot," said Moyes.
"We have a game in hand over Liverpool, so we will see what points we can pick up."
Until Gibson's fortunate opener gave the home side the lead QPR were arguably the better team, but offered little after going behind.
"I was worried beforehand about the game," Moyes added.
"They have really good individuals that are capable of hurting you, but I thought we handled them well.
"It was far from easy, and we had to keep playing, but we were well worthy of second goal, and the win in the end."