He then laid one on for team-mate Jonathan Franks in first-half stoppage time.
Substitute Leroy Lita added a third after the break, while home debutant Jordon Mutch treated fans to another special strike in the closing stages.
But aptly it was Johnson who wrapped things up a the death, this time with a free-kick.
The first real chance came as early as the fifth minute when James Spicer could only sting the palms of Middlesbrough goalkeeper Danny Coyne as the angle tightened.
Having been kept relatively quiet during the first 20 minutes, Boro settled, but for all their possession they struggled to create any notable chances.
That is, until Rovers defender James Chambers did magnificently to get himself in the way of a Johnson shot which looked almost certain to end up in the back of the net.
The hosts again went close when Billy Sharp almost beat Coyne from outside the box in the 25th minute, but the former Wales international saved well.
Then a moment of genius from Johnson five minutes later following a quick throw-in on the left gave the visitors the lead.
The 22-year-old winger was allowed an inch of space and he punished the onlooking Rovers defenders by thumping a left-footed drive into the top right corner of Neil Sullivan's goal.
And then in first-half stoppage time Boro doubled their lead when Franks scored a well-taken goal.
Johnson, who accelerated towards the byline with devastating speed, pulled the ball back for Franks who made no mistake from close range.
A third came in calamitous fashion when a mix-up between James O'Connor and Sullivan allowed Lita to score with 24 minutes still to play.
O'Connor had tried to nod the ball back to his keeper but it sailed over him, presenting Lita with the simple task of knocking in to an empty net.
Another sensational strike in the 84th minute gave the home supporters something to cheer.
Teenage midfielder Mutch marked his arrival on loan from Birmingham with a shot which flew past Coyne from 30 yards.
But it proved little more than a consolation as some quick thinking from Johnson in stoppage time saw him stroke home his second of the evening.
Rovers were slow to assemble any sort of wall after conceding a free-kick and the impressive youngster made them pay.