Los Angeles Galaxy striker Keane, who overtook goalkeeper Shay Given as Ireland's most capped player, turned in Aiden McGeady's cross to give Ireland the lead after five minutes, the highlight of an otherwise uninspiring half for the hosts.
Ireland began the second period just as lethargically until Keane, 33 next month, netted again 10 minutes after the break following good work from Wes Hoolahan. He rounded off the haul with his 59th international goal 10 minutes from time.
Ireland are level in second place in Group C with Austria, 2-1 winners over Sweden on Friday. After a 6-1 thrashing at home by leaders Germany earlier in the campaign, the Irish have just a plus-two goal difference compared to Austria's 10.
Sweden, who trail the second-placed sides by three points, have played a game less than both. (Reporting by Padraic Halpin, editing by Ed Osmond)