Stoke boss Michael O'Neill called for his side to be more clinical in front of goal despite booking a place in the last 16 of the Carabao Cup.
But the hosts had other chances to put the contest out of reach and did not take them.
"We dominated the game from start to finish, that's a very pleasing performance," reflected O'Neill.
"A lot of our play was excellent, but we need to be better in the final third. My ears are bleeding after that [match] though their bench were screaming for everything.
"But we dominated the game and clearly deserved to win it. It's early in the season, but tonight was a positive for a lot of the players."
Campbell's first goal since the start of July secured a first home win over the Gills for 16 years.
And Northern Irishman O'Neill was delighted his blossoming young striker is off and running for the season.
"It was a well-taken goal," said O'Neill. "It's good for any striker to be finishing. He [Tyrese] always knows what he's going to do. It was a good finish.
"If you're a striker then the ambition is to score lots of goals, I'd be worried if it was single figures he was aiming for."
Substitute Jordan Brown netted late on from a corner, but the effort was disallowed by referee David Webb, a decision which baffled O'Neill.
"I had no idea why Jordan's goal was disallowed, it was the clearest goal you'll see," said O'Neill. "It was an incredible decision!
"But it's another clean sheet, it will build [keeper] Adam' [Davies] confidence. We looked very assured at the back.
"The lads that played tonight gave me things to think about for Saturday [away to Preston]. That's what we want, the young players are really pushing the other players.
"The league is very difficult to call this season, they'll be a lot of teams with the aspiration to challenge for a top-six this season.
"Preston are a very settled side, it will be a tough test for us."
Frustrated Gillingham boss Steve Evans admitted his club were stretched to field a team.
"We did really well in the first half in terms of organisation and shape, especially as we came here with limited resources," he explained.
"The lowest you could get to be honest. We're here with bare bones.
"It was hard for us, but we showed real discipline. However, we gave a poor goal away.
"We said keep our discipline in the second half and we might get that one chance that comes, but we never really created it.
"We had to stay in the cup tie for 75-80 minutes, but we just never made that cutting edge chance."
Evans added: "We said we're going to go young this season and supporters have said they're heard that before from managers.
"But we're going young with quality. You look at that back four tonight and see our quality.
"They're kids, it's like a kindergarten when I walk into the changing room. But I'm really proud of them.
"It was a big test for us because we've got so much youth, but I've said to the boys be very proud of what you've done here tonight. But we've lost."