Ally McCoist's side beat Championship rivals Hibernian in the first round of the competition before Hearts were dumped 4-1 by Livingston this week.
And Hateley hopes the silverware has Rangers' name on it this season.
He said: "Alistair is keen to win this trophy and every time Rangers enter a competition they want to come out on top. That's our mentality and Alistair has always been a winner, even if he has been playing a game of dominoes.
"For me, I'd have said Hearts were the favourites, although Rangers would obviously have been up there as well.
"Now that they're out, it's a body blow for them and a real boost for everybody else who is still involved.
"Everybody wants a home draw but going to East Fife on a nice playing surface against a young side isn't the worst game for Rangers to get. It could have been tougher."