The Serie A champions were made to work hard for their slender victory though, secured after Ricardo Carvalho brought down Alvaro Morata in the box, with Vidal thumping home the resulting spot kick 12 minutes after half-time at the Juventus Stadium.
And Juve will now be favourites to progress to a first semi-final appearance for 12 years in Europe???s premier club competition when the two teams meet again in the return leg a week on Wednesday.
However, after an expected bright opening from the home side in which Morata volleyed over the bar from just six yards after being picked out by an inch-perfect pass Andrea Pirlo pass, it was actually Monaco who slowly began take control of proceedings.
First Yannick Carrasco passed up a golden opportunity to give the visitors a vital away goal to take back with them to France after being picked out in the area following a fine driving run by Anthony Martial, only for the Belgium winger to somehow fail to beat Gianluigi Buffon from close range when it seemed easier to score.
Then just seconds later the same player again tested Buffon with a low shot from the edge of the area, which the Juve captain did well to turn round the post after the ball was deflected en route to goal.
And when an unmarked Layvin Kurzawa also directed a header off target just past the quarter-hour mark, the home fans??? unhappiness with how Monaco were dominating was clear for all to hear.
But Juventus soon recovered their poise, and control of the contest, despite strong Monaco appeals for a penalty after Giorgio Chiellini appeared to haul the speedy Martial down in the box, with the hosts missing a glut of opportunities as the game moved towards the break.
First Tevez failed to make good enough contact with the ball after being found unmarked at the back post by Claudio Marchisio???s curling left-wing centre, before left-back Patrice Evra blazed wildly over the crossbar after fine approach play on the right-hand by-line from Roberto Pereyra.
However, it was Vidal who was guilty of the worst miss of all after the Chilean was played clean through on goal by yet another delicious defence-splitting pass from Pirlo, only for the midfielder to miss the target with only Danijel Subasic to beat in the Monaco goal.
The second period was dominated by Juventus, who began the half on the front foot after surely being on the receiving end of a half-time telling-off by coach Max Allegri, and it did not take them long to make the breakthrough.
Once again veteran midfielder Pirlo, who was making his first start after seven weeks out with a calf injury, was given too much time to pick out a devastating pass for Morata to run on to and as the Spain international prepared to shoot, Monaco centre-back Carvalho took his legs from under him.
The man who mattered decided it was a penalty, and the Portuguese was lucky to only be booked after seemingly denying Morata a clear goalscoring opportunity, although Juve were not complaining after Vidal then sent Subasic the wrong way from the spot.
Monaco reacted to falling behind by introducing mercurial frontman Dimitar Berbatov with 18 minutes remaining, and the Bulgarian very nearly drew his side level with his very first touch, only for his glancing header to just drift over the bar.
Thereafter Juve comfortably saw out the closing stages, but they will be far from confident of sealing the deal next week given they have lost every Champions League contest they have ever played on French soil.