There was confusion as to who scored the 89th minute goal, with Dejan Lovren credited by some as grabbing it.
However, Lallana has said that he got the final touch and claimed that the team are now building "great momentum".
"My apologies to Dejan, but it was my goal, I am claiming it," he told Sky Sports News.
"It come off my knee, but look it doesn't matter who scores, it was a great point. We rode our luck at times, but we also had chances, and we could have nicked it at the end. Great credit to the lads.
"Could we have won it? Definitely. We had chances at the end, but we are building great momentum. Credit to the lads, and the fans, we're going back home with a great point."
The draw leaves the Saints inside the top four in the Premier League.