West Ham sit fifth in the Barclays Premier League heading into Saturday's visit of struggling Crystal Palace and Slaven Bilic's side also face an FA Cup quarter-final replay at home to Manchester United later in the month to secure a place in the final four.
Winning the cup and qualifying for the Champions League would be the perfect finale to West Ham's last season at their Upton Park home - and would usher in their new era at the Olympic Stadium in style.
And Lanzini, who joined the club on loan from Al Jazira in the summer and has impressed enough to agree a permanent deal starting in the summer, reckons both targets are achievable.
He said: "I've heard a lot of people talk about whether they'd prefer to win the FA Cup or qualify for the Champions League. I say both, why not?
"That's how strong I believe our squad is. We have to believe we can do it; we have been optimistic all season and look where we are. We're in the latter stages of the FA Cup and close to the Champions League spots, so we have to go for it."
While hoping for success on two fronts at the end of the campaign, the 23-year-old former River Plate youngster is taking nothing for granted at this stage.
"We are very happy with our progress so far," Lanzini added.
"But we have to keep our focus, we haven't achieved anything yet. Each remaining game is huge, and we will fight, as we have been, to try and win every game.
"Palace haven't been having the greatest run of results recently, but that can happen to any team and we know Palace are a very dangerous side.
"They are very quick, and like to attack so we have to be aware of that. They switch the ball very quickly so we will have to be alert to deal with their qualities."
And Lanzini, who scored on his Hammers debut in their 2-1 defeat to Astra Giurgiu in Europa League qualifying, is looking forward to continuing his West Ham story after agreeing a permanent move.
"Of course, I am very happy to have signed for West Ham on a permanent deal, I feel happy, proud and excited," he said.
"I'm very grateful to my team-mates, coaching staff and the club for all their help so far in welcoming me and helping me to adapt to a new team and a new city."
Source: PA