Fox's head of domestic distribution, Chris Aronson, characterized the opening as "a very promising start," noting that the film won't open in Europe until the fall.
The Adam Sandler produced Sony comedy Grown Ups 2, starring Kevin James, Chris Rock, David Spade and Salma Hayek, held onto fourth place in its second week of release, earning $20 million.
Summit Entertainment's Red 2, which stars Bruce Willis and Helen Mirren as retired CIA operatives, debuted in fifth with $18.5 million.
But Universal's big-budget crime caper R.I.P.D. opened with a disappointing $12.76 million. The film reportedly cost more than $130 million to make.
The Jeff Bridges-Ryan Reynolds picture joins other big-budget summer flops such as Pacific Rim, White House Down and The Lone Ranger.
Overall box office totals are up, though, said Paul Dergarabedian of box-office tracker Hollywood.com.
"Despite the fact there have been some high-profile, high-budget flops, the summer to date is up 12 percent from last year," he said. "For mid-July, to have a modestly budgeted horror film top the weekend tells you how important it is for audiences to have options."
The Conjuring was originally set for release in the winter, but audience tests proved so positive that the studio moved the film to the heat of summer movie season.
"It was really a bold choice, no question about it," said Jeff Goldstein, Warner Bros executive vice president of theatrical distribution.
"With three other movies against us this week with high price tags and high expectations associated with them, this result is even more impressive."
He said the film cost about $19.5 million to make.
Still, there was no chance for this weekend's movies to match box-office totals for the same weekend last summer, when The Dark Knight Rises opened with $161 million domestically.
Estimated ticket sales for Friday through Sunday at U.S. and Canadian theaters, according to Hollywood.com. Final domestic figures will be released on Monday.
1. The Conjuring, $41.5 million.
2. Despicable Me 2, $25 million.
3. Turbo, $21.5 million.
4. Grown Ups 2, $20 million.
5. Red 2, $18.5 million.
6. Pacific Rim, $15.95 million.
7. R.I.P.D., $12.76 million.
8. The Heat, $9.3 million.
9. World War Z, $5.2 million.
10. Monsters University, $5 million.
- AP