Scotland bounced back from their disappointing opening-day defeat by England to sweep to a 34-10 victory over an Italian side that had stunned France last weekend.
Tim Visser notched his fifth try in seven internationals as Scotland controlled the first half without ever looking in danger.
And they wrapped the game up within six minutes of the restart as Matt Scott crossed for his first international try and Stuart Hogg ran almost the full length of the field to score after intercepting a pass, with Italy set for a walk-in score.
Sean Lamont also got a breakaway try before Alessandro Zanni's late consolation helped the Italians into double figures.
Interim head coach Johnson said: "We are really happy with the scoreboard and who wouldn't be? But we are in a championship and we want to win the championship.
"Whilst there were some really good things in that game and much improvement in areas we needed to improve in, there is still plenty left out on that pitch."
The Australian added: "We did really well in the defence for most of the game and there was a real turning point when Hoggy took it from one end of the field to the other, which was really a 14-point turnaround.
"But the events that occurred just beforehand were disappointing. The scoreboard could have changed really quickly.
"We have to acknowledge that but I keep saying if we get our part right, we can put sides away from some distance because we have a potent back line.
"We just have to keep doing our work. We have to keep improving. We're in a tournament and today's result puts us right in it. We're in this."
Johnson had been bitterly disappointed with his side's physicality and defence in their 38-18 defeat at Twickenham and told his players in some frank team meetings.
And a tackle count of 146 with a 92% success rate showed they had responded well during spells of Italian pressure in the second half.
"There is no point telling a kid without being completely honest," said Johnson, whose team next host Ireland in two weeks.
"There is such a propensity in the world when you are in trouble that you try to go to the end before you understand the beginning. We get to the end pretty good when we get the opportunities.
"The reality is we have to do a lot of the beginning. It's a reckoning that this is what we need to do and do it well.
"The contact and collision area is such a big part of the game and we are deluding ourselves if we think otherwise. There were periods today when we showed there is no excuse why we can't do it. We have got the physical specimens to do it and if we get it right, rugby is far more enjoyable."
Hogg's try was the perfect example of turning defence into attack as he intercepted Luciano Orquera's pass to prevent Italy scoring before streaking away himself.
The full-back said: "Fortunately the ball came right into my hands and it just opened up after that. I was really happy to run the full length and score and get points on the board for the team.
"I think it was (Martin) Castrogiovanni, the prop, (who was chasing me) so I would be quite disappointed if I didn't beat him. It opened up and I thought 'I'm going for it now' and I was fortunate enough to get there."
Scotland captain Kelly Brown was delighted with the fight shown by his players as they won their first Six Nations game for almost two years.
"We said last week after England that they won all the one-on-ones but it felt out there that we fought hard and worked hard as a side," Brown said.
"My overriding emotion is pride. We knew after last week we needed to bounce back and I felt we did that.
"But it's only the start. It's only one game and it's up to us to make sure we learn from this match and on our next outing are even stronger."
Italy coach Jacques Brunel was disappointed with his side's defending but brushed aside claims he had taken too many risks with his tactics.
"I think that's the price we pay to keep on growing," he said. "Today we certainly paid a high price.
"The Scots defended really well and there were many issues with our speed and defence throughout the game as well.
"We played in such an individual way, now we have two weeks to try to understand and come back for the Wales game."