The Blues ran out comfortable winners thanks to two second-half goals, one being an own goal from Gaetan Bong and the other a curled effort from Cenk Tosun but they had chances to further extend their lead.
Speaking to the media after the game, Allardyce said: “I think that the quality told in the end. We were only missing final third quality in the first-half to get in front after dominating it. After we produced that quality, we got ourselves in front - deservedly so.”
The Blues were aided by a number of good performances across the pitch, with Allardyce picking out Wayne Rooney as the Blues’ key man.
He said: “It was a shame Wayne missed it (the penalty) because he had a very good performance in the sitting midfield role. While all the players played really well, it’s the first time he’s played in that position and I thought he mastered it - he was our prompter.
“He kept the ball moving and kept the flow of the attacks going and overall, its a very good performance which isn’t unusual at home,” Allardyce added.
“A well-deserved result against a team who haven’t lost in seven and beat Arsenal in their last game.”
Had Everton scored in the first-half, there could have been little complaints from the visitors as they struggled to get a foothold in the game until the dying embers of the first-half.
When the teams emerged for the second period, it was again the Blues who dominated proceedings and ultimately found their lead.
On the goals, Allardyce stated: “I think that Yannick’s (Bolasie) ball was a shame that it’s an own goal and not a Theo (Walcott) goal, but the pressure Theo put them under caused the defender to put the ball in his own net.
“The second goal was why we bought Cenk into the club. That’s two in two now. We saw the type of finish that he delivered, at the right time for us, because going into the final stages - one nil up can be very nervous.”
Gylfi Sigurdsson picked up a knock on the knee during the clash but was able to see the game out.
Walcott picked up, what looked like a serious injury, but only has cramp and should be ok. Allardyce also added that Bolasie came through unscathed and picked up some much-needed match sharpness.
Idrissa Gana Gueye, who was a late change in the line-up, picked up a bug and missed proceedings through illness.