Coventry loanee Moore fired the visitors into an 81st-minute lead at the Priestfield Stadium before summer signing Connolly bagged a second goal to stun the home side.
And to rub salt into the wound, there was still time for Pressley to bag a third in added time as the Saddlers started the League Two season in style.
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Midfielder Moore, 20, pounced with a sweet strike to stun the home side and, as the Gills looked to get a leveller, they were punished again as former Crewe defender Connolly tapped home from close range.
Pressley then found space to score with a header in what turned out to be a brilliant finale for the Saddlers.
Walsall might have won, but the visitors were indebted to goalkeeper Jed Ward who made a string of fine saves to deny Ronan Hale, with James Brophy and Will Goodwin striking the bar for Gareth Ainsworth's side before they conceded a late flurry of goals.
