After their 8-0 pummeling of San Marino on Friday, along with Montenegro’s 1-0 win in Moldova, England sit two points behind Tuesday's opponents, with the reverse fixture at Wembley still to come.
And Cleverley claims Roy Hodgson's team will play to win in Podgorica in an attempt to take maximum points from the two-game international break.
"It is a significant game but we are going out to win it," the 23-year-old told reporters. "Two wins from these two matches would be really satisfying.
|11/1||Tom Cleverley is 11/1 with Bet365 to score first against Montenegro
Despite only claiming independence from Serbia in 2006, Montenegro have competed strongly in recent years and held England to a 2-2 draw the last time the sides met in qualifying for Euro 2012.
After thrashing minnows San Marino on Friday, there have been suggestions that a step up in class may cause England problems but Cleverley has shaken off any such notions.
"I don't think the adjustment from one type of opposition to the other will be that difficult," he added.
"We know the job we have to do. As long as we all work hard and play our football in the right way, we can get the job done."
Cleverley only made his debut for the national side in August of 2012 but he has not looked back since, appearing in every game for the Three Lions in their current qualification campaign.
"I have only had a short international career but I have played in some big games, against Brazil and Italy and the World Cup qualifiers," the Manchester United playmaker said.
"I have had some good experiences in my short time with England. Hopefully Tuesday will be another."