Despite this trouncing of a relegated Laois side in Cashel, Tipperary have to settle for a place in the Division 1 quarter-finals.
Level on points with Tipp at Division 1B's summit, Galway are through to the semi-finals thanks to their better scoring average.
Tipp led Laois by 3-07 to 0-05 at half-time with goals after six, ten and 17 minutes from Lar Corbett.
Tommy Fitzgerald fired home Laois' only goal but Tipp's own Tommy Fitzgerald, on as a substitute, then netted at the other end.
This was the first NHL fixture to be played at Cashel's Leahy Park since the 1983/84 season when Tipperary scored a 3-15 to 2-08 victory over Antrim.
Liam Sheedy's current crop knew they needed to win and win well today in order to have a chance to overhaul Galway and secure a direct spot in the semi-finals.
In freezing conditions, they made an excellent start with the hard-working Pat Kerwick slamming over a first-minute, which James Young replied to with a '65'.
Corbett then latched onto a long through to sweep home his first goal, six minutes in.
The Laois defence was caught badly napping for the goal. They had to made a switch before the throw-in when injury forced Brian Campion out of the game.
Joe Phelan reverted to full-back with Joe Dooley being added to the midlanders' attack.
Tipp also made a late alteration with former All Star Eoin Kelly ruled out due to a back injury and Seamus Butler taking his place in the starting line-up.
It made little difference to the Tipp attack as they struck 2-06 in a devastating 12-minute spell.
Points from Shane McGrath, Conor O'Mahony, Darragh Hickey, Joe Ceasar and Shane Maher sandwiched Corbett's second and third goals and saw the home side canter clear.
O'Mahony's score was a superbly accurate shot from 90 metres out, while Corbett's 17th-minute goal saw him tap home after Laois goalkeeper Tadhg Dornan has saved the initial shot.
A shellshocked Laois mustered their first point from play in the 26th-minute when Willie Hyland fired over to make it 3-07 to 0-04 - Young had landed three early placed balls.
Full-forward Hyland got off the mark again in the final minutes of the first half and while Tipp were 11 points to the good at the break, their management will have been unhappy with a wides tally of eight.
Two points from John Rowney and Zane Keenan cut the deficit for Laois on the resumption but Tipperary quickly added to their tally through Butler, with his second point a brilliant 70-metre effort off his left.
The hosts continued to press for scores with Corbett, Kerwick and Butler again on target.
In response, Young notched a point from play and a free but the gulf in quality was all too obvious.
Fitzgerald did add some respectability to the score-line with a goal on 60 minutes but Tipp, despite hearing over the PA system that Galway were beating Limerick well, kept their foot on the gas.
Points from Pa Bourke (0-02) and Butler were followed up by substitute Fitzgerald's goal, which saw him scramble the ball home after a Hickey shot had been saved.
