The midfielder claimed an impressive double on his competitive debut, with Ian Black and David Templeton also on the scoresheet in the Ramsdens Cup first-round tie.
And Law insists the Scottish League One side are capable of making an impact in all competitions this term - including the League Cup and the Scottish Cup.
He said: "We feel we are strong this year and we feel we can compete on all four fronts, not just in this competition and the league.
"It was just about going out there and putting a marker down and letting teams look at that and think that Rangers mean business this season.
"I thought we played well and probably could have won the game by more than four goals in the end."
A transfer embargo means Rangers cannot register their new recruits until September 1 and Law and Jon Daly both featured as trialists against Rovers.
The former Motherwell man said: "On a personal note, to score the two goals made it a perfect debut and I thoroughly enjoyed it.
"I'd just like to say thank you to the fans for the reception they gave me and for singing my name. It was brilliant.
"Hopefully they have left the ground feeling positive - not just about myself but about the whole team and the season ahead."
Law is now hoping for a bigger goal haul in the coming season compared to previous campaigns with Motherwell.
He said: "Double figures is what I'm looking for. I think I got seven and six in my last two seasons at Motherwell, which was okay.
"Now I'm at Rangers, I think I'll probably get a few more chances. I'm certainly looking for double figures and this was a good start."
Rangers struggled to adapt to their new surroundings in the bottom tier last season but Rovers boss James Ward expects them to be much stronger this term.
He said: "I think they recognised what they went into last year and now they have signed the players who can handle the situation.
"Jon Daly will be an absolute menace to people with the balls in the box and Nicky Law's quality and movement is excellent.
"They've got players around them like Ian Black and they've got a bit of flair here and there so they've got a better balance and I think it's a more settled environment for them as well."