The Dark Blues responded to their mortifying 20-0 drubbing by the Auld Enemy by securing a narrow 21-20 RBS 6 Nations victory in Rome.
Duncan Weir's last-gasp drop goal secured the win at the Stadio Olimpico but Lawson claims it was the pain inflicted on them by Stuart Lancaster's men that was the real key ingredient to their success.
He told Press Association Sport: "You have to enjoy winning when you are representing your country.
"But we have to use the pain of those first two defeats to drive us on. We did that against Italy and we need to remember what that feeling is like because that is what will make sure we don't suffer it again.
"This is now the benchmark. We have put in a winning performance against Italy in Rome, which historically has been a hard place for us to win.
"It is a step forward but we have France at Murrayfield in two weeks and it would be great to take another step forward by beating them too. But we will have to step it up again.
"We don't just want to be the team that can beat Italy. Yeah, it was great and an amazing feeling when the final whistle went, b ut we want to win the next game and then the one after that."
The Scots have had it hard in this year's championship, losing comprehensively to Ireland in Dublin before England steamrollered them at Murrayfield.
That defeat prompted a fortnight of soul-searching and recriminations in the press, with former players and pundits queuing up to slate the current Dark Blues line-up.
Former Twickenham boss Sir Clive Woodward even claimed it was the worst Scotland side he had ever seen but Lawson insists the display in Rome was a truer reflection of what the side are capable of.
"Ultimately we were all relieved when Duncan scored his drop goal," he said. "We showed a lot of character to get ourselves back in the game after half-time when we were 13-3 down. We showed a lot of resolve.
"Looking back at the first two games of the championship I don't think we have done ourselves justice. But we had that type of performance in us.
"We have had a lot of criticism and rightly so. But we backed ourselves for 80 minutes against the Italians and got our reward."
Lawson refused to swipe back at Scotland's doubters, though, admitting the flak was merited.
"We've lost games back-to-back before and it's not been a big deal," he said. "But it was the way we lost those first two that was different.
"The Scottish public is right to be aggrieved about that.
"But even if we had not got that last-minute drop goal on Saturday, you could not have questioned the boys attitude and commitment to the cause."
Lawson started his first match of the Six Nations in the Eternal City as he replaced the misfiring Ross Ford.
The Edinburgh man was dropped as a result of his erratic line-out deliveries but Lawson proved a safer pair of hands as Scotland won a perfect 10 out of 10 from their own throw.
"It was good," said the modest Newcastle forward. "When things like that become an issue you have got to address them.
"We worked hard on it this week and had a look at it. The results speak for themselves."
Scotland recorded their first tries of the tournament thanks to Alex Dunbar's second-half double and head coach Scott Johnson has tipped the Glasgow centre to form a midfield partnership with Matt Scott that will rival anything in the northern hemisphere.
Lawson refused to disagree and said: "Johnno has taken guys like Duncan Weir, Matt Scott and Alex Dunbar and decided to expose them to Test-match rugby, and the pressure that comes with that, as much as he can.
"That will stand all of them in good stead.
"The forwards gave them a better platform to work off against Italy and allowed them to go and play. They showed how good they are. It's about learning but these guys are doing well."