Back in the tournament after a 20-year absence, the Azzurre top Group C with six points ahead of Brazil and Australia who have three. Jamaica have yet to earn a point.
Girelli opened the scoring with a re-taken, VAR-awarded penalty before netting again either side of the interval.
Galli rounded off the victory with two goals after coming on midway through the second half.
Schools from around Reims brought their students to the game, giving the atmosphere an upbeat feel.
Three days after the U.S. thrashed Thailand 13-0, the Stade Auguste Delaune crowd were treated to more champagne football.
Italy went ahead after 12 minutes when Girelli converted a penalty on her second attempt, having had to re-take her spot kick after keeper Sydney Schneider had stepped off her line.
The penalty was awarded following a VAR review after Allyson Swaby brought down Barbara Bonansea.
Girelli doubled the tally in the 25th by latching onto a corner before securing her hat-trick shortly after the restart, jumping higher than Schneider to head home.
Galli added the fourth shortly after coming on for Valentina Bergamaschi with a thunderous 25-metre strike in the 71st.
Ten minutes later, she rounded Schneider before tapping the ball into the net to wrap up an impressive win.
Italy next play Brazil with Jamaica facing Australia.
