Lawrence headed home in the 17th minute while Britton settled the outcome six minutes into the second half.
The Swans had Rory Fallon in the starting line-up after he passed a late fitness test, but Jody Morris and Darren Byfield, both long-term absentees, were once again missing for the Lions.
The home defence was under pressure as early as the second minute when a first-time shot from Danny Haynes struck the woodwork, with Willy Gueret beaten.
Lee Trundle had Lions keeper Lenny Pidgeley scrambling to save a low shot after he outwitted Richard Shaw and cut inside.
A free-for-all on the touchline in the 11th minute led to Alan Tate and Tony Craig receiving yellow cards from referee Bill Jocelyn.
The Swans took the lead in the 17th minute when Millwall failed to clear a corner and Lawrence was perfectly placed to head in from close range.
Trundle's trickery almost set up a second for Swansea as his cross into the danger area was narrowly missed by Andy Robinson's slide at the far post.
There was a second flare-up in the 41st minute with Haynes involved, but once again only a yellow card was produced by the referee.
Millwall made a change for the start of the second half with Mark Phillips replacing Craig in the back line.
Six minutes after the re-start the Swans deservedly went further ahead when the Lions failed to clear and Britton, running in at the far post, slotted the loose ball into the net.
Willie Donachie threw his other two substitutes on in the 55th minute in an effort to get his side back into the game.
The Swans were now well on top and Owain Tudur-Jones had a goal-bound header brilliantly pushed away by Pidgeley.
On the hour, the busy Trundle shook off a tackle by Shaw and raced half the length of the pitch before being brought down by Derek McInnes. He took the spot-kick himself, but Pidgeley dived to his right to push the ball away.
That miss seemed to take the sting out of the Swans' play and in the last quarter Millwall enjoyed their best spell without ever looking as though they could claw their way back into the match.