The Glasgow club had the upper hand throughout the majority of the first half, and scored the first goal of the game after just 12 minutes when Dean Shiels curled an effort from the right-hand edge of the Brechin box into the top corner.
Just after the restart, Daly nearly scored a third for the hosts, but the ball cannoned off a defender for a corner.
Within a couple of minutes, Brechin then pulled a goal back against the run of play when Steven Robb struck a powerful shot that flew into Rangers' top corner.
The home side's control on the game further slipped away from them on the hour mark, as Bilel Mohsni was given a second yellow card for kicking the ball away, reducing Rangers to 10 men.
However, the Glasgow outfit were able to hang on for the victory, ensuring that they remain 23 points clear at the top of the league.