Darcheville gave Rangers a 60th-minute lead and Whittaker clinched it in stoppage time with a wonderful solo effort.
The goals sent Rangers into the last four of a European competition for the first time in 36 years.
Walter Smith's team were under pressure at times but Allan McGregor hardly had a save to make as he was well protected by Carlos Cuellar and Christian Dailly.
The only disappointing note for Rangers was that skipper Barry Ferguson was booked late on for kicking the ball away and will miss the first leg of the semi-final.
Rangers made an assured start and Darcheville had an early chance when he was through on Rui Patricio but his shot was blocked by Gladstone.
But Sporting hit back and showed they were prepared to try their luck from distance although McGregor was untroubled in the opening 15 minutes.
But there was a major let-off for Rangers after 17 minutes when Liedson powered in a header from a free-kick which beat McGregor but crashed back off the post.
Neither goalkeeper was particularly stretched in a first half that ebbed and flowed but lacked spark in front of goal.
Torrential rain made conditions difficult for both teams but Sporting looked more dangerous as half-time approached.
Joao Moutinho lashed a shot narrowly wide when he cut in from the right with three minutes left with McGregor exposed.
Sporting started the second half strongly and put Rangers under the cosh and Kevin Thomson picked up a booking for a lunging tackle.
Simon Vukcevic flashed a header inches wide with McGregor struggling to cover his far post as a goal looked imminent.
But it came at the other end. Some slack defensive play by Gladstone gave Darcheville the chance to send Steve Davis clear and he threaded the ball back to the striker who slotted it home from ten yards.
It meant Sporting had to score twice to go through and Rangers kept McGregor well protected in goal
Yannick Djalo should have equalised with 14 minutes left but shot wide from 15 yards after the Rangers defence had been sliced open.
Tonel then sent a six-yard header marginally wide as Rangers lived dangerously in the closing stages.
Incredibly, Whittaker made it 2-0 in stoppage time when he burst from midfield and beat three defenders before slotting the ball home.