Laidlaw's late try and 26-point haul proved telling as the Scots overcame Samoa in a 36-33 thriller in Newcastle.
"Greig had an outstanding game. We were under a lot of pressure; he kept his cool, we kept our cool," said Cotter.
"We just needed to win, and that's done. We're relieved, but we're proud as well - that was a tough game."
The Scots leaked three tries to a dynamic, incisive Samoa during the furious first 40 minutes, losing Ryan Wilson to a yellow card for stamping, but responded with tries from Tommy Seymour and John Hardie to trail by three points at the interval.
The Pacific Islanders were already out of contention for a berth in the last eight, but Scotland needed a win to wrap up second place in Pool B.
"That was an outstanding performance from Samoa," Cotter reflected. "They've obviously spent this week having a look at how they want to play, and wanted to finish on a high.
"They threw the ball around, they've got really dangerous players. They were impressive. They had us under pressure, and we just had to stay in, keep believing.
"We knew in the second half we would perhaps get an edge, and finally we did."
Cotter's side will face Australia in the quarter-finals after they beat Wales 15-6 at Twickenham.
"It was about winning and fighting for a win," added the coach. "Because if anyone thought it was going to be an easy game, after the first five minutes, we certainly got a taste of what was coming at us.
"We found ways into the game and I thought the forwards stepped up and played well. I'm really, really happy for them."
Laidlaw joked that defence coach Matt Taylor "is a couple of years older after that game", but said there was no half-time panic.
"Defence was probably the hot topic at half-time," the scrum-half said. "But to be fair to the coaches, they were calm and controlled at half-time and that spread through to the players.
"We were only three points down and that was with a yellow card, so it was stick to the game plan, wear them down, and we did that."
Scotland head coach Vern Cotter hailed the performance of Greig Laidlaw after his side set up a World Cup quarter-final against Australia.