There have been only two wins and a draw since Johnson's men clinched the series against the then world champion Springboks 2-0. The last series against the Qantas Wallabies, in 2001, also ended in defeat for the first time in a three match series.
But Warburton believes the desire to make history, starting with a win in the opening Test in Brisbane on Saturday, will spur his side on to perform at their peak in the series.
"Everybody is desperate to achieve. It's the one thing you will see in the body language of every player for every game, because none of us has achieved a Lions Test series win," said Warburton.
"It's something you desperately want to have on your rugby CV. We've got Heineken Cup winners, Grand Slam winners, English champions - all these competitions, but nobody's ticked the Lions box.
"That's something which I think we will be pretty desperate in wanting to achieve. That's the one thing which will be a certainty in all three matches - the players' determination to get a result."
In Warburton's favour at the weekend will be the fact the Wallabies haven't played since December, 2012, and will be fielding three new caps. History also shows that the Lions have yet to lose a Test in Brisbane, with seven successive victories to date stretching back to 1899.
The Lions may have lost their unbeaten record against the Brumbies in Canberra on Tuesday night after five wins on the trot, but Warburton doesn't feel that will affect morale in any way.
"The Brumbies were a well drilled outfit - physically excellent, they competed hard at the breakdown. But we had five players in the back-line playing their first game," said Warburton.
"It was always going to be a difficult ask against the top side in Australia. It won't have affected our confidence - you don't have time to dwell. If we can win the Test series people won't remember the Brumbies game."
The Cardiff Blues flanker has led the Lions to two weekend wins, against Queensland Reds and the Waratahs, after overcoming a knee injury and will be looking for a hat-trick at Suncorp Stadium. Playing for the Lions in a Test match has been the biggest goal of his playing career and he can't wait for the experience.
"I have been thinking about this for quite a while, how it might feel. But I won't truly know until I actually do it," he added.
"The last two games have been awesome, but I know it will be at another level again on Saturday. I have led Wales out in some pretty big games, but that will be blown out of water by the Lions, which is the ultimate pinnacle in rugby - I'm very proud to be doing it."