Tommy Seymour and John Hardie each dotted down amid the madness, while the boot of Laidlaw - now the highest point-scorer of the tournament - kept them in the hunt.
Scotland's starting XV was their youngest ever at a World Cup and, defensively, that inexperience showed before half-time but they eventually began to assert more control.
The skipper crossed over late on and, despite a last-gasp Motu Matu'u try, Scotland - who began the day just two points ahead of third-placed Japan in Pool B - are safely into the last eight.
Pisi and Laidlaw traded penalties inside the opening ten minutes before a thrilling half was sparked into life by the game's opening try.
Alafoti Faosiliva took Samoa deep into Scottish territory and an overlap on the right allowed fly-half Pisi to stretch out and score in the corner. The TMO confirmed the try and its scorer converted expertly from the touchline.
Scotland were suddenly level, however, and it was a gift, Seymour capitalising on a loose pass just five metres out to juggle the ball and dot down, Laidlaw adding the extras.
The see-saw start continued and, with just 13 minutes on the clock, hooker Leiataua - making his World Cup debut - coasted over after a quickly-taken penalty from Northampton Saints scrum-half Kahn Fotuali'i wreaked havoc in a ragged Scottish defence.
The game's early pattern suggested Scotland would score next and they did, courtesy of a simple Laidlaw penalty on 20 minutes, but things soon got worse.
A third Laidlaw penalty reduced the gap to four but flanker Ryan Wilson was soon in the sin bin for a stamp and Pisi restored some breathing space from the tee.
Vern Cotter's charges hardly looked like a side battling a numerical disadvantage, tough, and Hardie was on hand to finish off a powerful rolling maul, Laidlaw's extras drawing them level.
Samoa thought they had a bonus-point try when Sakaria Taulafo piled over, only for an obstruction on Laidlaw to see it chalked off, but another Pisi penalty handed them a slim lead at the break.
After a collective deep breath it took 11 second-half minutes for the scoreline to change, when Laidlaw added the first of two quickfire penalties that put Scotland ahead for the first time.
The skipper then missed back-to-back efforts from the tee and Scotland went into the final ten minutes - when Sean Lamont was introduced for his 100th cap in place of British & Irish Lions Stuart Hogg - just 29-26 up.
WP Nel almost eased the nerves in Newcastle when he burrowed over following a spell of sustained pressure but the prop was held up by Fotuali'i.
They didn't have to wait long to make amends and it was Laidlaw sniping over from close range and converting five minutes from time.
Replacement front-rower Matu'u piled over three minutes later to put the pressure back on but Scotland held their nerve in the dying seconds to win it.