Wales' preparation for the game had been horribly compromised in a wretched week where the fixture was in doubt after the players threatened to strike in their row with the Welsh union over contracts and other issues.
Backed by the ever-emotional crowd, the hosts somehow remained in the fight, five points adrift, until the last five minutes when Lawrence's try secured a deserved win for England, their first in Cardiff since 2017.
Ireland, who beat Italy 34-20 earlier, top the standings on 15 points. Second-placed Scotland, who visit France on Sunday, have 10 points along with England. Wales remain rooted to the bottom without a point.
"I thought it was a step forward for us. As we keep saying we're at the start of our journey," said England captain Owen Farrell.
"It was a different challenge for us to come away from home to a place like this, a team that has had a lot going on during the week and have unified to come together for this crowd at the weekend. Fair play to them.
"We had a job to do and thankfully we took a step forward.
"It's not all a matter of scoreboard, it's about building pressure as well. We deserved to be more up at times because I was bad off the tee today but if we'd have done that Wales would have had to force their hand a bit earlier."
England started full of positivity and, after an early Farrell penalty, scored an excellent try on a first-phase move from a scrum.
Number Eight Alex Dombrandt showed his skills with a crisp miss-pass to Watson, who finished well on his first England start after two injury-hit years.
Frustratingly for England they immediately conceded a penalty that Leigh Halfpenny slotted to give the Cardiff crowd something to get their teeth into.
England continued to look the more dangerous side without really threatening the tryline but also defended strongly, not least when the home side had 20 phases hammering around the 22 as Wales continued their championship theme of failing to turn pressure into points.
Farrell then missed a penalty as England went in 8-3 ahead at halftime but within a minute of the restart Wales were in front after Louis Rees-Zammit intercepted a Max Malins pass and Halfpenny converted for a 10-8 lead.
England then kicked a penalty to the corner and were rewarded when the forwards forced Sinckler over with Farrell converting for a five-point lead.
Both sides, seemingly more desperate to avoid defeat than push on for victory, struggled to hold on to the ball as the second half drifted into a succession of tactical kicks and the tension in the stadium was palpable.
Farrell missed a 45-metre penalty shot after an hour but England remained the dominant side and crucially pulled clear five minutes from the end when Lawrence showed great determination to blast over.
"Really proud of the team - that was a tough test with a lot of pressure," said England coach Steve Borthwick.
"Wales deserve enormous respect after the week they've had. I thought our intention to move the ball to the edge was good but the Welsh jacklers slowed us down."
England next host France while Wales visit Italy, both on March 11.