The Ibrox side are still waiting for their first away win in the Irn-Bru Third Division, having also shared the points at Peterhead on the opening day of the league campaign.
Brechin also ran McCoist's men close in the Ramsdens Cup, with an extra-time winner required to progress, and the manager insists his side should have seen enough by now to realise exactly what they will be up against this season.
He said: "It's a fact, every away game is going to be a Scottish Cup tie. That's the way it is, that's what we are going to be faced with.
"Teams will be up for it, they will be aggressive, they will be in your face. They will also try and play, it's not kick and rush.
"It's something we are going to be faced with and they've got to realise that - they must realise it.
"They've had four away games now and every one of them has been the same in the respect that the opposition has rolled up their sleeves, shown no respect and got in our faces.
"That's the way we would expect it but, at this moment in time, we are not reacting to it too well."
Asked if his players have an attitude problem, McCoist added: "That's probably as big a criticism as you could level at any player so I would be loathe to say that.
"We've certainly got problems in terms of the way we are playing because that was miles short.
"It looked as though there was a lack of confidence, a lack of sharpness."
The day started well for Rangers when Andy Little netted his seventh goal of the season to open the scoring on the stroke of half-time.
Berwick hauled themselves back into the game thanks to Fraser McLaren, before Chris Townsley was unfortunate to have a last-gasp goal disallowed for a foul in the build-up.
McCoist said: "Ian [Little] should be proud of his team because they fought and got at least what they deserved.
"But, having said that, I've got to look at it from our point of view and there were very few, if any, aspects of our performance that I could give pass marks to."
Referee Mike Tumilty ruled that goalscorer McLaren fouled goalkeeper Neil Alexander at the corner before Townsley netted what would have been a famous winner.
McLaren said: "It was very disappointing. I was just standing my ground and he ran into me and keepers get everything, don't they?
"We've just got to deal with it. It should have been 2-1 but we'll take the draw. We would have taken that before the game."