Arturo Vidal's first-leg penalty was enough to take the Serie A champions into the semi-finals, although they suffered a few scares at the Stade Louis II.
Massimiliano Allegri's defence remained resolute to keep a clean sheet and progress, with Evra making no apologies for the style of their performance against his former club.
"We must give a nod to Monaco, who deserved more tonight. But from our point of view, we never really doubted ourselves," he told beIN Sports.
"We played 'Italian' - it's solid, it's ugly, but it happens.
"I am happy and at the same time a little sad for Monaco. It was strange to return to Louis II, where I spent many happy years."
Gianluigi Buffon was required most urgently to save from Geoffrey Kondogbia, and also mopped up a loose ball with Dimitar Berbatov set to break on goal.
With Juve back in the competition's semi-finals for the first time since 2003, the veteran goalkeeper told Sport Mediaset: "I thought I'd have to wait less when I was at this stage 12 years ago.
"Of course, we know that we haven't achieved anything yet. We wanted to get through to the next round at all costs.
"We knew it was a great opportunity and after 12 years we wanted to honour the history of Juve."
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