Replacement number eight Billy Vunipola's stoppage-time score, confirmed after a three-minute TMO review, was England's fourth try and gave them five points in the standings.
With the potential for Australia, Wales and England to beat each other in a tough Pool A, a bonus point could prove key in deciding progress to the quarter-finals.
"If we been offered four tries and a bonus point I would have absolutely taken it," Lancaster told reporters after Vunipola added to two tries for Mike Brown and an early penalty try.
"We had to work hard and that last try late in the game was a massive relief. It’s a big try in the context of this pool. Bonus points are absolutely invaluable, it puts pressure on sides to go out and get four tries against Fiji," he added.
"Even at 18-11, I don’t think they broke us down defensively and I was confident our fitness would see us through."
It was a mixed night for England, who started really well amid a tingling atmosphere but lost their way against a physical and determined Fijian side before finishing strongly.
"We didn't get our game plan into action. We have to be more direct, we were a little bit lateral. Our bench made us much quicker," said Lancaster, who threw on six replacements around the hour mark which gave the team an immediate uplift.
"I felt we needed some energy. We have a very strong bench and we can trust them to come on and make a difference," he said.
"We had to trust the game would come to us in the last 20 minutes."
With the result in the bag and eight days to work with before facing Wales, Lancaster allowed himself to enjoy the night.
"The whole thing was phenomenal as an occasion for the players," he said. "Twickenham was rocking when we arrived and the boys were like caged animals in the changing room."
Captain Chris Robshaw, who was again to the fore when England's late forward drives drained the last reserves of energy from the islanders, was also delighted.
"We knew it would be a tough test, all the emotion of the week, and think the guys handled the huge sideshow well," he said.
"All in all to get five points we are in a happy place.
"We knew it would come down to last few minutes, though not last few seconds."
(Editing by Ed Osmond)
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