Substitute Lallana struck five minutes from time to cancel out Rashford’s first-half opener and deny the Red Devils a first win in six games in all competitions.
It was a much-improved performance from United, who were beaten 1-0 at Newcastle before the international break.
That is of little consolation to Rashford, though, who felt his side had done more than enough to earn victory.
"We deserved the three points,” he told Sky Sports . “I don't think they were good enough to get a draw out of the game or a win but this is football.
“It was a big opportunity missed and we just fell short. Hopefully in midweek we can get a win and it will start a run for us.”
United boss Ole Gunnar Solskjaer responded to that Newcastle defeat by tinkering with his formation and playing Rashford and Daniel James as split strikers.
The tactic worked perfectly in the first half, with James teeing up Rashford for the opening goal, though the England international played down the significance of the positional switch.
"It wasn't a big change to what we were doing at the start of the season,” he added. “On paper it’s split strikers but it’s pretty much the same thing. It was a bit more back to normal.
“It was good to have that stability but I'm disappointed we didn't get the three points.”
There was some controversy about Rashford’s opener, with Liverpool claiming it should have been disallowed due to a foul by Victor Lindelof on Divock Origi in the build-up.
The goal was eventually given following a VAR check, leaving Liverpool enraged, but Rashford admits he was not aware of the dispute.
“I didn't know what was going on to be honest,” said the 21-year-old. “I didn't see anything in the build-up as to why it wouldn't be a goal.
"There have been other games where we have been very unlucky. It was fine margins.
“We can play a lot better, especially in attack. There is still a lot of work to do but I think we should win this game today."