The 20-year-old, deputising for the injured Stevie Hammell at left-back, twice bulleted corners into the roof of the net after clever runs across the home defence.
Motherwell survived some difficult moments and Darren Randolph came to their rescue on several occasions, but they were able to progress while resting Steve Jennings, Jamie Murphy and John Sutton from their starting line-up in their 12th competitive match of the season.
Craig Brown also handed Alan Gow his first start for the club in the main striker's role while the recalled Ross Forbes played just behind him in a 4-2-3-1 formation.
Brechin had Rory McAllister back after suspension and the lone striker carved out an early chance when he turned Page and cut the ball across the box for Calum Booth, but Keith Lasley made a crucial sliding challenge.
Motherwell, who won 3-1 at the same venue in a pre-season friendly, soon settled and Page headed over from eight yards when he ran onto Forbes' outswinging corner.
Brechin failed to heed the warning and the defender soon headed firmly into the roof of the net from an identical move in the 16th minute.
However, the goal did not herald a period of superiority for Motherwell and the hosts created a great chance in the 26th minute.
Craig Molloy headed a Motherwell clearance into the box to set McAllister free, but Randolph stood up well and parried the striker's effort.
City midfielder Neil Janczyk also came close when his rising shot flashed just wide from 30 yards following good build-up play.
Motherwell moved to a 4-4-2 formation for the second half but they survived two major scares in the opening four minutes.
Jamie Redman created both chances from the right wing, with his first cross landing at Booth's feet after a misjudgement by Mark Reynolds, but Randolph turned the Hibernian loan midfielder's shot behind his near post.
Redman then skinned Page before substitute David McKenna knocked his cross towards the far post where McAllister prodded it wide from two yards at full stretch.
McAllister lobbed over with an audacious effort as Brechin remained on top, but Nick Blackman almost doubled Motherwell's lead with a brilliant 25-yard strike which Craig Nelson acrobatically pushed over.
Motherwell soon made Brechin pay for their missed chances when City again failed to match Page's run as he flashed Tom Hateley's near-post corner home from close range in the 59th minute.
Nelson impressively denied Gow his first Motherwell goal by tipping the former Falkirk player's shot over his top corner after a slick break.
But the Second Division side refused to wilt and continued to force Motherwell back for significant spells and Randolph again showed his agility to turn Molloy's flick wide.
The Brechin fans knew their luck was out though when McKenna turned McAllister's low cross onto the underside of the bar and saw it bounce on the line and into Randolph's arms.