Goals in either half from Offaly's Ger Oakley were not enough as Laois held on to claim their first NHL points at O'Moore Park.
Laois were full value for a 1-10 to 1-06 interval lead with their goal coming from half-forward Eoin Holohan in the 26th-minute.
The hosts opened up a seven-point advantage on the restart but were made battle until the end, with Oakley, Brian Carroll and Joe Bergin in fine form.
However, an injury-time point from James Young sealed the Laois win.
Despite a strong second-half surge from Offaly, Laois held out by a slender margin in mixed weather conditions in Portlaoise.
The 3,000-strong crowd were treated to strong sun, rain and a prevailing wind throughout.
Laois goal-scorer Holohan, along with free-taker Young, helped pave the way for a home victory but this was a resolute team performance, with two sideline cuts from Zane Keenan the most notable scores of and end-to-end contest.
Laois were quickest out of the blocks and after 20 seconds, they had a point on the board from Holohan.
Young (0-03), Joe Dooley and Zane Keenan also found the target quickly to hand the hosts a 0-06 to 0-01 lead.
Offaly then got their first score for 17 minutes with Carroll on target, but this was a mere blip in a dominant spell from Damien Fox's charges.
Keenan restored Laois' five-point lead, following a well-hit sideline cut.
Sheer hunger and energy were two qualities which Laois brought in abundance to this game and when Holohan found the net after a Tommy Fitzgerald pass, they rushed into a 1-8 to 0-3 lead.
Offaly gradually came back and with Oakley hitting the back of the Laois net a minute before the break, just four points separated the sides at half-time.
Laois were full value for a 1-10 to 1-06 interval lead, although Offaly did cause them problems in the middle third.
Diarmuid Horan floated in a through ball for Oakley to notch his second goal in the 42nd-minute.
Overs from Carroll, Bergin and Shane Dooley kept Offaly in the hunt but Young's sixth point of the afternoon was enough for Laois to claim both match points.
LAOIS: P Mullaney; J A Delaney, B Campion, C Dunne; J Phelan, M Whelan, C Coonan (0-01); J Fitzpatrick (0-01), J Walsh; J Young (0-06, 0-03f), E Holohan (1-02), Z Keenan (0-03, 0-02 sl); T Fitzgerald (0-03), J Dooley (0-01), N Holmes.
Subs used: W Hyland for Dooley (44 mins), J Brophy for Holmes (64), S Norton for Walsh (69).
OFFALY: S O'Connor; C Hernon, K Brady, K Grogan; D Horan, J Bergin (0-03, 0-01f, 0-01 '65'), D Kenny; C Mahon, J Rigney (0-01); S Dooley (0-02), G Oakley (2-01), B Carroll (0-05, 0-03f); D Currams, D Molloy, D Franks (0-01).
Subs used: B Teehan for Currams (6 mins), G Healion for Grogan (38), J Brady for Kenny (47), F Kerrigan for Mahon (62).
Referee: Cathal McAllister (Cork).
