"With respect to Leigh, I don't want anyone thinking we are signing a Premier League striker, because that's not the case," said McCarthy.
"It's a big step up for Leigh, but he's got a terrific goalscoring record."
Scotland Under-21 international Griffiths has netted 29 times in 56 appearances for Scottish First Division side Dundee, scoring 10 goals this season.
"What we shouldn't take our eye off is what we have always done," added McCarthy.
"And that is sign players who we think might be valuable to us and be a bargain.
I don't expect Wolves to start competing with sides who are happy and prepared to spend upwards of £20m on one player, but at least compete with the teams on a similar level
"We've done it in the past, the Wards, Foleys, Edwards, Kightlys, you can name lots and lots of them we have done shrewd deals for and they have been good and smart deals."
The Edinburgh-born striker began his career at Livingston before joining Dundee at the start of last season. He is due to sign a two-and-half-year deal at Wolves.
Griffiths will be the club's second acquisition in the space of a week, following Adam Hammill's arrival from Barnsley for an undisclosed fee on 20 January.
Source: BBC Sport