Ryan Fraser scored twice to earn the hosts the win.
“It is a frustrating way for the unbeaten run to end because we should have avoided both goals, that is the bottom line," said Allardyce, who had presided over a sequence of seven undefeated matches as Toffees boss prior to the game at Vitality Stadium.
“You cannot go away from home and gift them two goals like we did today. We put ourselves in a good position to come back from 1-0 down and had good possession in the opposition half.
“But we just gave it away and they went down the other end without anyone really shutting them down. They got a shot off, and a bit of luck, but we should never have let it get to that position."
The manager continued: "You look at the goals today and they could both easily have been avoided – the first one particularly, which was a sloppy pass. That will happen. But we should have defended that better from then on, rather than the man [Fraser] being in our box free to score.
“It was very disappointing to lose our unbeaten run against a side that has been struggling this season.
“We could see that [Bournemouth had been struggling], because once we did move the ball quickly and in the right way, and got through the lines, we did create some chances.
“I thought we would get at least a draw when we scored in the second half because we were looking a lot better as an attacking force. But the way we lost the two goals was a big disappointment."