Chiefs feel pressure after losing perfect Super Rugby record
Replacement fly-half Tom Lynagh landed a 78th-minute penalty for the Reds in New Plymouth on Friday night to seal one of the shock results of the season -- one the visitors say was built on defending "like a pack of dogs".
The Chiefs are still guaranteed to finish this weekend atop the standings regardless of the remaining matches after winning all of their first 10 games.
But McMillan said there will be extra pressure on them over the final three weeks of the regular season, starting with next Saturday's home match against a strong Wellington Hurricanes.
"We felt no pressure from the winning streak, we didn't even really talk about it," McMillan told broadcaster Sky TV.
"But there's no doubt this brings a little bit of pressure now, it brings us closer back to the pack and that might be a good thing too.
"We'll sharpen the sword a little bit and look forward to the last few weeks."
McMillan conceded his team failed to match the desperation of a Reds side that notched their fifth win to climb just inside the top eight.
He believed the Queenslanders' superior long-kicking game was a key factor and that other teams will have taken note.
"They did a good job shutting down our threats and I think they won the kick battle, so we spent a lot of time at the wrong end of the field," McMillan said.
"Other teams will see that, so if there's a positive, we'll start finding things out about shortcomings in our game now.
"They looked like a desperate side and more desperate than us. That's disappointing but perhaps a good reality check."
Reds captain Tate McDermott said his team's victory was encapsulated by the closing minutes, when the Chiefs launched an attack that went through 27 breakdown phases.
The visitors held firm on their try line, including the final act, when Chiefs hooker Tyrone Thompson was held up.
"We've always just lacked the polish, and discipline has been an Achilles heel for us," McDermott said.
"The boys kicked and defended really well, like a pack of dogs, so I'm so proud of that."