"I thought we played very well tonight. The pitch was excellent and that was reflected in the fact we were passing the ball better," said Smith.
Rangers survived an early scare when Liedson headed against the post and Vucevic and Veloso also went close before half-time.
While Sporting dominated much of the possession during the second half, they failed to test goalkeeper Alan McGregor and Darcheville capped a powerful performance to turn the tie into Rangers' favour. In injury time, Whittaker jinked his way through the heart of the Sporting defence and kept his cool to seal the victory.
"We got a break when they hit the post in the first half, but once we got in front we looked good.
"There are one or two celebrations in the dressing room and deservedly so."