The Slovaks are set to make their first trip as an independent nation to the European Championship after finishing second in Group C behind leaders Spain, who beat third-placed Ukraine 1-0 to end their hopes of an automatic berth.
Ukraine, who finished with 19 points, three behind Slovakia (22), will now feature in next month's playoffs.
European champions Spain topped the group with 27 points from their 10 games.
Slovakia coach Jan Kozak's side were anxious to put behind them the struggles of their previous three matches, in which they took only one point and failed to score after starting the qualifying campaign by reeling off six straight wins.
The team squandered a chance last Friday to wrap up its Euro 2016 spot but lost 1-0 to group minnows Belarus. However, Kozak said he never worried.
"I never doubted (qualification) for a second," he told Slovak news website SME.sk after the match.
"This team has character, quality, and I was convinced we could do it ... To advance to the European Championships is something fantastic."
Leading scorer Hamsik ended Slovakia's scoring drought in the 24th minute, collecting a Vladimir Weiss pass on the right wing and launching a left-footed shot into the net.
Weiss assisted again five minutes later, placing the ball in the front of the goal where Adam Nemec tapped it in. Moments later, Weiss completed a trio of assists, this time finding Robert Mak alone in front of goal for another easy tap-in.
Hosts Luxembourg, though, battled back in the second half when Mario Mutsch's punt sailed over the keeper's head in the 61st minute. Lars Gerson then scored with a penalty a few minutes later.
But that was as close as Luxembourg would get as Slovakia's defence prevented an equaliser.
Hamsik then capped the scoring for Slovakia with a goal in added time, striking home Stansislav Sestak's pass in the box to the delight of a vocal legion of Slovak fans that had made the trip.
"We achieved another historic success for Slovak football and we are just happy we can be a part of it," Hamsik said.
(Reporting by Jason Hovet; Editing by Ken Ferris)
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