Both teams scored three tries in driving rain on Friday night, Damian McKenzie's perfect five-from-five kicking the difference after the Reds were gritty in the first half but made to pay for missed opportunities early in the second.
Les Kiss's side arrived as top of the Super charts for the first time in 13 years and showed why in a defensive masterclass against a side that had won 19 of their last 21 games at FMG Stadium.
Repelling everything that was thrown their way, the Reds were unlucky not to lead at halftime after impressive backrower Joe Brial kicked a loose ball 40 metres ahead to score before the buzzer.
Both of Harry McLaughlin-Phillips' conversion attempts just missed to leave it 10-10 at the break.
Question marks over Brial's grounding were waved away by the television match official, who was thrust into the spotlight again after the break when McLaughlin-Phillips beat the line and was awarded the try as he slid onto the line.
But the five-pointer was reversed when the TMO adjudged the flyhalf to have lost the ball before reaching the line.
The Reds won the scrum against the feed from the restart and, with prop Jeff Toomaga-Allen off the field after being yellow-carded for high contact in the ruck, opted for a lineout.
Their maul broke down though and the Chiefs took the points when presented the option at the other end, McKenzie slotting a penalty to putting his side ahead 13-10.
The Reds' resilience continued with key defensive plays from Brial, Fraser McReight, Harry Wilson and Filipo Daugunu to stall the host's charge.
But eventually their line was breached for a second time - Luke Jacobson had scored their first try after just three minutes - when replacement hooker Samisoni Taukei'aho rolled over behind a maul.
Xavier Roe's try iced the contest for the hosts but Wilson showed his class with a left-footed grubber to gift Tom Lynagh a try in injury time.
"We were heroic at times in defence in holding them out," Reds coach Kiss said.
"Some moments maybe that we regret ... a couple of things like that we didn't clean up tidily enough but the effort … very pleasing, very proud."
Matt Faessler scored the Reds' first try while scrumhalf Tate McDermott was again impressive in difficult conditions that saw the Chiefs struggle under the high ball in the first half.
"It was a sloppy scene out there," Jacobson said.
"We've just got to be clinical. It's tough in weather like this but you don't make excuses and there was some awesome play out there tonight so we've got to be happy."
McDermott said the fact his side were a man down played into his decision to kick for touch rather than take the penalty at 10-10.
"If we had 15 players on the field we would take the three there," he said.
"I'll wear that as the captain; we backed the boys, that's what we've done all year.
"It didn't come off and we'll review why."
The Reds must regroup ahead of a home clash with the Brumbies next Saturday.
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