Two of Laws' 11 summer signings were on the scoresheet with a goal each before half-time.
Midfielder Matt Sparrow celebrated returning to his hometown team for a second spell by opening the scoring after 29 minutes.
Former Wolves striker Sam Winnall made it 2-0 10 minutes later with a debut goal for the club.
"An opening-day game against a newly-promoted team is always the hardest you can get," said Laws.
"There is a momentum and a belief about them and Mansfield showed that they are going to do all right in League Two this season.
"After being relegated ourselves last season, of course it was vital for us to get off to a good start in front of our own fans.
"We have brought in 11 new faces this summer and they are beginning to believe in what we are trying to achieve.
"I knew if we defended our box well against the aerial bombardment that Mansfield threw at us, then we would be all right if we could then get the ball down ourselves and play.
"We showed we can do that, we showed how we are going to approach things in League Two this season.
"But we are not going to get carried away. This is just one win - one that before the game we craved for.
"The question now is can we go and build on that."
Laws was delighted with the contribution of Sparrow, who spent 10 years at Scunthorpe and helped them win promotion from League Two to Championship level.
"His goal was important for us because it settled us down. I thought he was excellent for us. It's brilliant to have him back."