Head coach Eddie Jones said England had used up all of their "get-out-of-jail-free cards", after Elliot Daly's 76th-minute try secured a 21-16 Six Nations victory over Wales in Cardiff.
That followed a 19-16 win over France in their opening match, when the winning try came in the 71st minute.
"We don't want to be in that position again," said Jones.
"We are a gritty team with characters in there that don't know how to get beaten, and that was evident here."
England, who have won a national record 16 Tests in a row, play Italy next.
The defending champions are yet to secure a bonus point in their first two games, and Jones said he wanted to "put Italy to the cleaners" at Twickenham in a fortnight's time.
After Ben Youngs' early try for England, Liam Williams' slicing first-half try and 11 points from the boot of Leigh Halfpenny looked to have given Wales a deserved victory.
But Owen Farrell's penalties had kept them within two points, and with time running out his long flat pass put Daly away down the left to score.
Jones said the match-winner - who features in the centres for Wasps - was being deployed in a position that suited the team rather than the player.
"The boy's got gas and he's got that X-factor about him and that's what we like about him," Jones said.
"I don't necessarily think wing is his best position, but it suits us at the moment."
The Australian also returned to a topic that had featured heavily in the build-up to the match - the Principality Stadium roof.
Jones used the away team's veto to frustrate Wales' wishes and keep the match open to the elements.
England have now won five out of six matches at the ground with the roof open, and lost four out of five when it has been closed.
"They can close the roof now," he said. "The roof should be open unless the conditions are going to be absolutely terrible. That's how rugby should be played because it's a winter sport, so you play the conditions."
Captain Dylan Hartley, who was replaced by Saracens' Jamie George after 47 minutes, paid tribute to the influence of England's bench.
"I would have preferred to wrap it up a bit earlier. The finishers came on for us and showed great composure," he said.
How did the pundits view it?
Former England hooker Brian Moore: "It shows again that if you do not put this England side away when you are on top they will make you pay.
"They were outplayed for long periods but when it came down to taking the opportunity from a poor Welsh kick, they found a way to win."
Former Wales fly-half Jonathan Davies: "I felt that England looked far more threatening with ball in hand. When the opportunity came, they took it.
"They were so clinical in the opportunities they had. Wales had a lot of possession, a lot of territory and scored a great try in the first half, but unfortunately they couldn't turn that pressure into points."
Former England scrum-half Matt Dawson: "I've never watched an England side with only 40-60% territory, under that much pressure, win a game. They didn't even nick it, they worked it.
"That game was absolutely superb. On that evidence, there is no gap now between the southern hemisphere teams."
Social media reaction
|2 (1)||Scrums won (lost)||7 (0)|
|8 (1)||Line-outs won (lost)||9 (0)|
|122 (2)||Rucks/mauls won (lost)||113 (5)|
|19||Kicks from hand||24|
|181 (16)||Tackles made (missed)||127 (18)|