Although the Tykes managed two scores, the scoreline flattered the visitors somewhat, and Leeds coach Phil Davies will be travelling home with a lot of thinking to do.
In Gordon Ross and scrum-half Alan Dickens, the Tykes sported a new half-back pairing, but such was the dominance of the Leicester forwards, the duo barely saw the ball.
Johnson rumbled over for the first try after 12 minutes, and Geordan Murphy's incisive burst set up debutant Seru Rabeni for Leicester's second just before half-time.
Home fly-half Andy Goode enjoyed a superlative day with the boot, and converted both tries and added two penalties, one from five metres inside his own half.
Leeds finally scrambled onto the scoreboard courtesy of Gordon Ross' boot in first half injury time, and a sensational opportunist try by Tom Biggs at the start of the second half could have had the Tykes fans dreaming of a comeback.
Goode soon put pay to any such dreams with another penalty, and when Leeds lock Chris Murphy was yellow-carded shortly after, the writing was on the wall.
Leicester scrum-half Harry Ellis then sniped over from a scrum, and just four minutes later, Leicester pressured a penalty try out of the Leeds forwards, who collapsed a maul on the line. Goode converted both scores to earn his side an unassailable 37-13 lead.
Henry Tuilagi slithered into the corner for Leicester's fifth five minutes from the end, and Jon Dunbar's consolation try for Leeds deep into injury time had only cosmetic value for the scoreline.
For Leicester Tigers:Tries: Johnson, Rabeni, Ellis, Tuilagi, penalty tryCons: Goode 4Pens: Goode 3
For Leeds Tykes:Tries: Biggs, DunbarCons: Ross 2Pens: Ross 2