Mercedes boss Toto Wolff says his team now has "no breathing space" in this year's championship given the threat from Red Bull.
Daniel Ricciardo beat the Mercedes drivers to pole position - Red Bull's first since the 2013 season - and looked set for victory before a strategic error allowed Lewis Hamilton to take the lead.
Red Bull benefited from the introduction of the upgraded Renault engine, and both the team and its rivals expect the improvement to be even more significant on other tracks.
Wolff labelled Red Bull's threat as "very real", and says Mercedes is under more pressure than before.
"The main thing we took away was the very real threat from Red Bull," said Wolff. "It took a bold strategy, a big push from Lewis and an even bigger slice of luck with Daniel's slow pitstop to get us that win.
"I've said this many times before - but we have no breathing space in this championship. Maintaining any advantage is a constant battle - and the pressure is only getting bigger.
"This unpredictable season has been great for the sport and shows that Formula 1 is alive and well. But, for us as a team, there is clearly work to do.
"There remains room for improvement in our reliability - but the push for more performance has become increasingly important too."
Hamilton "as strong as ever"
Hamilton secured his first win since 2015 in Monaco, as teammate Nico Rosberg had to settle for seventh position.
The British driver said his Monte Carlo victory showed he is still at his very best, despite a difficult patch at the start of the year.
"I've proven that I'm just as strong as I've ever been and I will be for the rest of the year, so I'm looking forward to the next chapter," he said.
"Montreal has always been a good track for me, so hopefully I'm able to shine like I did the first time I went there in that great city atmosphere."