Cork was dropped to the substitutes’ bench for the game, after the return from injury of Victor Wanyama.
Given his performances in recent weeks, it might well have been a tough pill for the midfielder to swallow.
But 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.
“Schneiderlin upped his game, myself, Jay-Rod, Lambo, Davo, Shaw, who was getting forward, and Clyne came on and had an impact.
“Boruc kept us in it in the first half and I think it was a real good team display. A result like that has been coming, looking at how we played at Sunderland and against Arsenal.
“It is well-deserved and credit to everyone at the club.”
With the three teams above them – Newcastle, Manchester United and Tottenham – all dropping points at the weekend, Lallana says Saints still feel they can climb the table and even get into Europe.
“We’ve got Newcastle in our sights now,” he said. “People were saying we were in our own little mini-league, but that has never been the case for us.
“We want to fight to win every game and finish as high as up in the league as possible. There are 14 games left and we want to be looking to win the majority of them and sneak into Europe.”