Cork was thrown on at the interval, and helped Saints overwhelm the Cottagers and race to an impressive 3-0 win.
"It was a bit frustrating in the first half," said Lallana. "We weren't at our best.
"It is a tough place to come, you can't just stamp your authority on it from the off. But, in the second half, I think every player stepped up. We said at half-time we needed to step it up an extra 20 per cent.
"I thought Cork coming in, for me, changed the game.
"I thought he was absolutely outstanding."