Fleetwood's three-under total for the day left him in a four-strong group on 11 under, one shot behind Bud Cauley and two off clubhouse leader Jackson Suber.
Suber signed for a 66 after a booming 348-yard drive set him up for a birdie at the par-five last and earned him the outright lead going into Sunday at a PGA Tour event for the first time.
Ben James, the pro debutant who ended Friday as the shock leader, came back to earth with a resounding bump as an eight-over 78 took him completely out of contention.
- Latest leaderboard: PGA Tour - RBC Canadian Open
- When are the majors? Key dates and 2026 results
- Get Sky Sports or stream golf with no contract
Fleetwood began the day two behind James and could scarcely have wished for a better start as he reached the green at the par-five first in two and holed a 16-footer for eagle.
That was as good as it got on the front nine, but a short approach at the 12th left him an eight-footer for a birdie, and another followed from five feet at the short14th after a fine tee shot.
The only blemish came at the tough 17th, where Fleetwood's second found rough to the right of the green and he failed to get up and down in two from there.
Only three players birdied the 521-yard par four all day, among them Billy Horschel, who did so with a putt from just shy of 99 feet on his way to a 64 and a total of 10 under.
Four others join him there, including Sam Burns, while Fleetwood's companions a shot further ahead are the Swede Jesper Svensson and American pair Brice Garnett and Wyndham Clark, whose seven-under 63 - featuring seven birdies and no bogeys - was the round of the day.
Watch the final round of the RBC Canadian Open on Sunday live on Sky Sports Golf with early coverage starting from 12.45pm. Get Sky Sports or stream with no contract on NOW.
