The Chile international clinically dispatched his seventh goal of the season in all competitions in the 57th minute, although Monaco claimed that Ricardo Carvalho's foul on Alvaro Morata had occurred outside the area.
Andrea Pirlo made his first Juve appearance since suffering a calf problem against Borussia Dortmund in the previous round back in February, and the playmaker had a hand in the winner.
It was his lofted pass from deep which Morata looked likely to latch onto until he stumbled under contact from Carvalho, with replays suggesting the initial foul had taken place outside the box.
Vidal's successful penalty was a welcome relief, after the midfielder had earlier wasted a gilt-edged chance to break the deadlock at the end of the first half.
Carlos Tevez was also guilty of profligacy prior to Vidal's winner, while Monaco - although still well in the tie - may rue Yannick Ferreira-Carrasco's inability to convert from close range in the opening stages.
As it is, Leonardo Jardim's men will go into next week's return leg at Stade Louis II needing to overturn a one-goal deficit to reach the last four of this competition for the first time since 2004, when they were runners-up.
Juve started the brighter, but survived a real scare in the 10th minute when Ferreira-Carrasco fired straight at Gianluigi Buffon from 10 yards after being picked out by Anthony Martial.
Claudio Marchisio's cross from the left picked out Tevez unmarked at the far post, but the Argentinian striker's volley six yards out was weak so Danijel Subasic was able to make a comfortable save shortly before the half-hour.
Monaco felt they should have had a penalty when Martial went down under a challenge from Giorgio Chiellini six minutes before the interval, but referee Pavel Kralovec was uninterested in their appeals.
Juve wasted another golden opportunity on the stroke of half-time, when Vidal blazed over after getting on the end of Tevez's superb ball with the outside of his right foot.
Monaco suffered a blow at the start of the second half when Nabil Dirar was forced off injured, and his replacement Bernardo Silva had an effort beaten away at his near post by Buffon.
Soon afterwards Juve took a deserved lead, Vidal sending a superb spot-kick into the top left-hand corner after Morata had been clipped by Carvalho.
Monaco responded and Buffon had to tip over Geoffrey Kondogbia's effort from range in the 63rd minute.
The remainder of the game proved largely uneventful with Juve seemingly content to protect their lead, leaving the tie firmly in the balance heading into the return leg in eight days' time.
