The runaway table-toppers were 2-0 up and cruising at half-time after another dominant opening 45 minutes, but Steven Robb pulled a goal back before Bilel Mohsni was sent off.
But Rangers safely negotiated the final half-hour with 10 men to retain their commanding 23-point lead over Dunfermline at the summit.
Brechin goalkeeper Graeme Smith was forced into action early on, saving a David Templeton shot with his legs before palming away a low effort from Dean Shiels.
Smith pulled off another high-quality save to keep out a close-range Jon Daly header, but Shiels then celebrated his 29th birthday with a stylish opener on 14 minutes.
Shiels cut in from the right and planted a delightful, curling effort beyond the leap of Smith and into the top corner, an effort which Templeton almost matched from the opposite flank moments later.
The barrage of chances continued when Nicky Law latched onto Daly’s pass only to drill a low shot too close to Smith.
But Daly made it 2-0 on 25 minutes as he gained a yard on his marker before leaping to glance Lee Wallace’s free-kick past the luckless Smith.
The Hedgemen goalkeeper did well to block another low strike by Shiels, and Daly nodded over the bar before Smith thwarted Shiels and Fraser Aird.
But the visitors improved after the interval and silenced the home faithful with a classy goal of their own, Steven Robb unleashing a thundering drive that fizzed past Cammy Bell and into the top corner.
Daly again headed over the bar as Rangers regrouped, and Smith made another sharp save from Wallace before Mohsni incurred a second caution on the hour for petulantly kicking the ball against Gerry McLaughlan.
Rangers continued to dominate despite their one-man disadvantage, and Smith again reacted well to foil Wallace at his near post before saving an angled drive from Daly.
Templeton and Law also went close, but Brechin almost stole an equaliser against the run of play when Andy Jackson’s header was cleared off the line by Richard Foster with Bell beaten.
The visitors finished strongly as they attempted to snatch a point, but Bell saved well from Jackson and then gathered a decent effort from Derek Carcary to secure the three points.