The Gunners trail Liverpool by seven points but have two games in hand, including Wednesday's trip to Southampton.
Liverpool have a difficult visit to West Ham and a home tie against an already-relegated Middlesbrough left to play.
"We have got to win the last two," Milner told British media. "How would we feel if we miss out? Sick. This is a team and a club that needs to be in the Champions League."
Liverpool have struggled for consistency in their last few games, losing at home to Crystal Palace, scraping a 1-0 win at Watford and drawing 0-0 at home to Southampton at the weekend in a game Milner failed to convert a penalty.
"We have shown on any one day we can beat anyone so if we could get in the Champions League we know we could be a match for anyone," he said. "We have got to do it consistently."
Milner added that Liverpool would be home and dry had they taken full points from sides down the table.
"We've dropped points where we shouldn't have done," he said. "Against the top teams, our record is very, very good but against the teams that come to sit back and play on the break and look to get something from a set-piece, we need to improve."
(Reporting by Aditi Prakash in Bengaluru; Editing by Peter Rutherford; peter.rutherford@thomsonreuters.com; +822 3704 5698 Reuters; Messaging: peter.rutherford.reuters.com@reuters.net)
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