Trailing 23-26 at the break, the Scots adjusted their tactics to shut down a Samoan side who had enjoyed nearly two-thirds of the first-half possession, run in three tries and had a fourth disallowed.
"We came across a Samoan team that we haven't seen in this World Cup. They were impressive, and we had to change the way we played," Cotter told reporters.
"Perhaps they did catch us unawares, but we still managed to stay in there and get the win," the New Zealander added.
"We needed to break down their way of playing, which was free-flowing and very physical. We had to hang on to the ball...so they couldn't get possession."
The switch of tactics worked, as the Scots slowed the frenetic pace of the game and pinned back the Pacific Islanders in their own half, forcing them to concede a series of penalties.
Captain Greig Laidlaw kicked two of them to pull Scotland ahead, and scored the decisive try when he nipped round the side of a scrum in front of the posts, having turned down the chance to kick for three points.
"Fortunately for myself I was able to dive over," Laidlaw told reporters after Scotland made up for the disappointment of falling at the pool stages in 2011.
"It's a great feeling, especially after missing out four years ago."
Laidlaw paid tribute to Cotter and his coaching team for steadying the ship during the break.
"They kept us calm... That was the message at halftime: don't panic, it will take 80 minutes to grind them down."
Laidlaw promised there was "more to come" from the Scots, who next play Australia or Wales.
"We're not finished," he said. "We'll be going into the quarter-final with some confidence. We believe when we get it right, we're a dangerous team."
(Editing by Ed Osmond)
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