Wolves put the result beyond doubt after a dominant first-half display when Griffiths, Lee Evans and Bakary Sako were all on target.
Scotland international Griffiths then grabbed a second-half penalty as he opened his account for the club after scoring 28 goals on-loan for Hibernian last season.
Jackett said: "We feel that if we can get our play right, get our passing right, and if we create, that Leigh Griffiths will put the finishing touches to it.
"If we can create chances like we did here then he is capable of putting those finishing touches to it.
"I'm sure he's delighted to get off the mark in his Wolves career in the first home game. Getting such an early goal in a game such as this is key, for both him and the side.
"I've found him fine in the month-and-a-half or two months that I've seen him. I got the job one day and phoned him the next and that's how I feel, having watched him last season.
"I told him 'You're not moving, you're coming back to Wolves and I'm going to give you a chance. You're going to play'. That's what he wanted, that's what every player wants and hopefully now that backing can be repaid."
It was Jackett's first win and he hopes a marker has been laid down. He added: "I've been looking for things to settle down now after basically pulling it apart.
"It was needed but now we need to try to get some continuity and for the side to just settle down and get a feel for each other.
"The supporters have wanted to see a new team, see some attacking play and some quality and, when needed, see the players roll their sleeves up and battle and scrap, if it's possible to have all those things.
"The group of players shouldn't get carried away but they should be encouraged by the standards they set and should also be able to build on that going forward."
It has been a real baptism of fire for Gillingham after they won promotion as League Two champions last season. Defeat at Wolves came after an opening day loss to Colchester and elimination from the Capital One Cup at the hands of Bristol City while they are also yet to score this season.
Manager Martin Allen said: "I thought it was very disappointing. I don't think we played anywhere near where we can.
"I can take a loss if there are some individual moments of brilliance but I thought we gifted them too many goals.
"All the players are very disappointed at not giving a good account of themselves on what was a big occasion for our club.
"The important thing is that we don't pop the pills when we lose and we don't pop champagne when we win so just take it all in your stride deal with it, face up to it and work at what you have to get better at and then move on to the next stage.
"The players need help. It is all very well thinking I have to shout at them. That is the last thing they want. You want someone to help and guide your through it.
"That is what we did last year and is what we will do. There is no panic. It doesn't get me down I have had this for 40 years so the knocks that set you back you deal with it and get on with it."