Gareth Southgate's men slumped to a 1-0 loss at Wembley in their final match before jetting off to Germany.
England, however, remain one of the pre-tournament favourites thanks to an abundance of attacking talent, featuring Harry Kane, Jude Bellingham, Phil Foden and Bukayo Saka.
They are expected to cruise through a group also featuring Denmark and Slovenia and Gallagher is hoping the Iceland performance will eradicate any danger of complacency.
"It's very important (to win the opening game) in tournament football," said the Chelsea midfielder.
"Just doing the first thing as well as you can - even if it is the right pass, winning your tackle, winning your headers, second balls, doing the basics - to give the team confidence to go and win that first game.
"It wasn't a good result and not the best performance (against Iceland). It could be good that it happened.
"It could be a wake-up call for us that it is not easy to win games against any country and we can be more than ready to beat Serbia."
Gallagher comes into the tournament with his club future still up in the air.
Despite captaining Chelsea for much of last season under Mauricio Pochettino, the Blues could look to ease their pressures to meet financial rules by cashing in one of the club's academy graduates.
Gallagher is reportedly attracting interest from Tottenham and Aston Villa.
Despite his insistence that his focus is fully on ending England's 58-year wait for major tournament glory, Gallagher was glowing in his praise for Pochettino, who left Chelsea are just one season in charge last month.
"I've come a long way this season. I give a lot of credit to Pochettino and what he’s done for me," added the 24-year-old.
"It puts me in a better position to be in the team this tournament.
"He just showed his belief and trust in me which is all a player wants. If a manager shows you a bit of love it makes you play better and, long story short, gives you confidence and that’s what he gave me."
England's preparations for the opening game were given a boost on Thursday as Manchester City defender John Stones returned to training after missing Wednesday's session due to illness.