The manager felt his side could well have taken all the points had things gone for them.
But Kidderminster came back into it after going behind and secured a 1-1 draw in the Vanarama National League North.
Bower said Harriers are sure to finish in a play-off place this year but said his team showed them they too can compete.
"We're certainly not getting any luck at the moment, but we stuck at our jobs and it's a decent point for us,” he said.
“It was a very even game and Kidderminster will certainly be in the top seven come the end of the season.
“We more than matched them and on another day we would have won the game. We stood up to Kidderminster so credit to the players.”
Avenue are just two points outside the play-off place while Harriers are in 12th.