A late surge, sparked by two tries from their young winger, Lee Smith, saw Leeds launch the defence of their Super League title with a precarious victory.
Rovers, who finished 11th in their first season among the elite last year, held their own for most of the game and only went home empty-handed because they were ultimately worn down by the Rhinos' pressure.
Despite defending an eight-point lead as though their lives depended on it, the Robins cracked when some typically expansive ball movement took play across field for Clinton Toopi's pass to allow Smith to squeeze in at the corner.
Eight minutes from time, Danny McGuire jabbed through a little kick near the Rovers' line, Brett Galea could not clean it up and Smith won the contest to get downward pressure on the ball.
Kevin Sinfield landed the conversion and was on the mark again to complete a thoroughly flattering final scoreline after McGuire got over with the last play of the night.
Rovers, with no less than nine newcomers in their team, had done enough to suggest that they are going to be hard to beat this year.
After Scott Donald had given Leeds the lead, it was one of those new boys who threatened to be Rovers' match-winner.
Jake Webster, the New Zealand international three-quarter, was a handful from the start and it was he that leveled the scores, finishing off a neat run-around move worked by Ben Cockayne and Clint Newton.
He also landed the penalty that gave the visitors a shock half-time lead after Gareth Ellis had strayed offside.
His finest moment, however, came when he raced half the length of the field for a glorious second try that gave his side a real chance to win at Headingley for the first time since 1985.
There were already some warning signs, though. Ali Lauitiiti had twice been within a finger-tip of opening up the Robins' defence with sublime offloads to Brent Webb, and tiring Hull KR players were conceding too many penalties.
"Our effort was excellent, but we didn't get our brains in gear for the full 80 minutes," said their coach, Justin Morgan. "We didn't have much possession in the second half and eventually we ran out of petrol."
Leeds' Brian McClennan knows his side will have to play a lot better to retain their title, but this was a tricky first match in charge, safely got out of the way.
"Our attack was pretty flat, but as the game went on we started to find our rhythm," he said..
Leeds: Webb, Smith, Toopi, Senior, Donald, McGuire, Burrow, Leuluai, Diskin, Peacock, Kirke, Ellis, Sinfield. Substitutes: Lauitiiti, Bailey, Scruton, Ablett.
Hull KR: Briscoe, Fox, Walker, Jake Webster, Fitzhenry, Chester, James Webster, Vella, Netherton, Mills, Galea, Newton, Murrell. Substitutes: Crossman, Gene, Aizue, Cockayne.
Referee: R Silverwood (Mirfield).