Graeme said in the Evening Times: "We have been branded inconsistent, so there is no way any of the boys has got carried away with the Munster win. All our focus has been applied on backing that up against Scarlets on Sunday.
"But there has been a bit of a buzz with the fans after the Munster game and we want to keep that going. It is not good enough that our last successive wins at Firhill were last autumn. So if there is one thing we must change it is an ability to win even when we are not at our best."
Graeme set up replacement fly-half Ruaridh Jackson for the decisive try, but the youngster dislocated a shoulder in the process and, after undergoing surgery, will be out for three months.
Graeme admits that injury has taken the shine off the opening-day success.
He said: "We are all gutted for Ruaridh but, unfortunately, injuries like that are part and parcel of the game and the good thing is that he has had his operation and is already now focusing on his rehab.
"But we have great cover with Colin Gregor able to come in and battle with Dan Parks for the stand-off position. That is the big difference this season in that we have competition for places in all positions on a level we never had before."
Graeme has had to form a new centre partnership with Peter Murchie after a cruciate ligament injury ruled out Max Evans for the start of the Magners League season.
"We have got on to the same wavelength pretty quickly and Peter handled himself very well against Munster and that is promising for him and for Glasgow," said Graeme.
Looking ahead to Sunday's match with the Scarlets Graeme thinks supporters will see a contrast in style to Munster's forward orientated game.
He said: "Scarlets have a lot of quality and they like to throw the ball about and then look for a channel. In the past we have maybe not defended that type of rugby too well, so we need to make sure that up front we don't take a step back and defensively the backs are water-tight.
"But we have given ourselves a platform with the Munster win and, if the pack can turn in the type of display they did last time out, then we have the quality in our backs to hurt any team."