Owen Farrell added 12 points with the boot before limping off with what appeared to be a groin or hamstring injury after an hour and his replacement Toby Flood slotted a brace of penalties to seal the success.
France arrived in London on the back of two tournament defeats in a row this term but they were always a threat after Philippe Saint Andre made seven changes to the team that was beaten by Wales in Paris in Round 2.
"It was a great win for us. It was a proper Test match," said Lancaster, who led England to second in the Championship in his first season in charge in 2012.
"That wasn't the same French team that played Wales a couple of weeks ago.
"They caused us lots of problems. It was hard work and it took 65-70 minutes to do it…but by the end we were the better side.
"We gave away a soft try on the blindside - we shouldn't be giving away tries like that at this level.
"Our composure grew in the second half. The substitutions made a big difference and helped us close the game out."
Farrell kicked England into a 6-3 lead but it was the visitors who were ahead at the break thanks to a fine try from Wesley Fofana. The Clermont Auvergne centre, who was played out of position on the wing against Italy and Wales, cut through from 40 metres as he transferred his imperious club form on to the international stage.
Courtney Lawes, Chris Ashton and Ben Youngs all failed to bring Fofana down after he picked up the ball two thirds of the way across the pitch and on his own 10-metre line and with seemingly little on ahead of him.
Morgan Parra added the difficult conversion to his fifth-minute penalty and France were four points in front after half an hour.
But Farrell cut the gap to a single point six minutes before the interval and the Saracens star kicked England 12-10 ahead just 60 seconds after the restart and the hosts never let their lead slip from then on in.
Tuilagi picked up a loose ball to power home for the decisive try from close to 40 metres out on 54 minutes, although England received a substantial slice of luck as referee Craig Joubert failed to spot the ball cannon off Mako Vunipola and into the arms of Tuilagi in an offside position.
Replacement fly-half Freddie Michalak reduced the arrears to just four points moments after Farrell failed with the conversion attempt but that was as close as the French got to just a second win at Twickenham since the tournament expanded to six teams.
Flood left France chasing a converted try with his first penalty success with eight minutes left to play and then put the game to bed four minutes later to ensure England march on towards a Round 4 meeting with Italy with a 100 per cent record and a two-point lead at the top of the table.