The visitors had already gone ahead with a 13th-minute header from Keith Andrews when their cause was further helped by the dismissal of Estonia defender Andrei Stepanov 11 minutes before the break.
Ireland made their man advantage count midway through the second half when Jon Walters headed in a second, four minutes before Keane's 71st minute strike put them firmly in control before Tuesday's second leg in Dublin.
Estonia's fading dream of reaching the finals of a major tournament for the first time was dealt another blow with captain Raio Piiroja's 77th minute dismissal and more indiscipline gave Ireland a late penalty which Keane converted.
"We came here to get a result but I don't think anyone would have thought it would be the result that it was," Keane told Sky Sports. "Full credit to the players, what a great performance."
Ireland's game plan was as simple as it was effective - an onslaught of high balls to Jon Walters up front, with Aiden McGeady, Keane and Damien Duff trying to pick up the breaking balls.
Walters got the goal his performance richly deserved, nodding home Keane's cross to double the Irish advantage in the 67th minute.
Soon afterwards Estonian goalkeeper Sergei Pareiko stretched out a foot to block Andrews' free-kick but Keane was on hand to slot home the rebound.
Ireland's record international goal-scorer Keane then scored his 53rd goal for his country when he fired in an 88th-minute penalty, given when substitute Stephen Hunt was hauled down, to make next week's return leg look academic.
Asked if the tie was effectively over, Keane replied: "It should be, the way we have performed here.
"Obviously we can't take anything for granted but if we don't qualify now we'll be devastated."
