The Brumbies, who were reduced to 14 men twice during the match, took what opportunities they had to score points while the Reds opted not to take several kickable shots at goal and were denied several tries through desperate Brumbies defence.
Brumbies inside centre Christian Lealiifano, who spent 10 minutes in the sinbin for a professional foul, scored all of his side's points with a converted try and four penalties to earn the draw that moved the Brumbies to 35 points in the standings.
"I'm disappointed," Reds captain James Horwill said. "We shot ourselves in the foot, we put them under pressure but just didn't execute to get over the line."
Lealiifano slotted two early penalties to give the visitors a 6-0 lead before the Reds refused kickable shots at goal to set up an attacking lineout from which number eight Jake Schatz crashed over from a rolling maul.
Brumbies full-back Jesse Mogg then gave his side field position with a 50-metre dash that sliced the Reds open and after recycling the ball for 11 phases to create an overlap, Lealiifano crossed in the corner.
He converted to give the Brumbies a 13-7 lead but was then sinbinned after the Reds had turned down two further shots at goal and taken quick tap penalties.
The home side, however, were unable add any further points even with the extra-man advantage and went into the break 13-7 down.
The Reds continued to attempt to play at a high pace in the second half with winger Rod Davies crossing in the corner after some superb build-up work from his forwards and following sustained pressure inside the Brumbies' 22-metre area.
Lealiifano slotted two additional penalties, but the visitors then spent the remainder of the match defending, with replacement prop Scott Sio sinbinned in the 69th minute.
This allowed Gill to score six minutes later and move the Reds to 33 points and fourth place in the table.
- Sports & Recreation
- Christian Lealiifano