The Italian champions travel to the Stade Louis II with a 1-0 lead from last week's first leg in Turin - Arturo Vidal's penalty settling a hard-fought match.
With the slender advantage, Massimiliano Allegri's side are favourites to progress to the semi-finals.
But Evra, who spent four years at Monaco between 2002 and 2006, stated that there is plenty of work to do and pointed to Monaco's spirited performance in the first leg as proof Juve cannot rest on their laurels.
"I was not surprised by Monaco in the first leg," he told Canal Plus. "They have not at all been underestimated.
"They gave a very good game. Against them [in the first leg] we were favourites. I think a lot of players felt that pressure.
"In the return leg, it will be different and we can take control of the game."
Monaco upset the odds to reach the last eight by beating Arsenal on away goals and Evra and believes they can pose a challenge for any side.
"When they won against Arsenal I was really proud of the team," he added. "They were fearless."
Juve, who defeated Monaco 6-4 on aggregate to reach the 1997-1998 Champions League final, make the trip boosted by a 2-0 victory over Lazio in Saturday's top-of-the-table clash in Serie A.
That result that left them 15 points clear at the summit in their march towards a fourth consecutive title.
By contrast, Leonardo Jardim's side have stuttered in recent weeks and a 1-1 draw against Rennes on Saturday made it three league matches without a win on home turf.
Third-placed Monaco are four points clear of Marseille and Saint-Etienne, while Lyon and Paris Saint-Germain are six points better off than the Principality club - with the champions also having a game in hand.
Head coach Jardim has stressed the importance of cutting out the errors against Juve.
"We lacked intensity at the end of the match [against Rennes]," he told reporters. "We started to feel fatigued. We'll recover and correct the small things for the match against Juventus."
Vidal will be assessed ahead of the game after missing training on Monday due to tonsillitis. Paul Pogba [hamstring] is unlikely to feature, while Juve team-mates Kwadwo Asamoah [knee], Martin Caceres [ankle], Luca Marrone [thigh] and Romulo [groin] are also absent.
Monaco are without Lacina Traore [leg], while Tiemoue Bakayoko [thigh] is a doubt.
