A first try for the club from Montoya, combined with the kicking of Zack Henry, gave Tigers a 13-3 advantage at half-time against a Warriors team who had enjoyed wins on three of their last four trips to Leicester.
And the hooker added a second, quickly followed by a first for the club by Wiese to stretch the lead.
Wiese and replacement Youngs added the other scores to take the full five points in Round 8 of the Gallagher Premiership season.
Richard Wigglesworth led out Tigers as captain in by his fifth appearance for the club and, after full-back Chris Pennell had been applauded on to the field by his team-mates in his 250th appearance, the game got under way on a bright but chilly winter afternoon.
Tigers dominated possession and territory in the early phases, but lost Nemani Nadolo to injury in their first attack when he stretched out for a crosskick from Zack Henry in the left-hand corner.
Henry had been playing with a penalty advantage and he knocked over the first points when play came back for the kick in front of the posts.
Argentina duo Julián Montoya and Tomás Lavanini were prominent ball-carriers in those exchanges, though replacement Kini Murimurivalu earned the praise of his team-mates when he smothered breakdown ball inside his own 22 on 15 minutes.
Untidy ball on halfway then led to a Warriors penalty and Duncan Weir hammered his kick between the posts to level the scores on 18 minutes.
Lavanini almost brought an instant reply, charging down a clearing kick just short of the Worcester line but then seeing the ball squirt out of reach.
Another carry from Montoya brought another penalty, and Henry pushed play into the right-hand corner. The forwards controlled play up to the tryline and were playing with advantage again for a high tackle on Dan Cole when a final push was held up over the chalk.
This time, after more strong work at the lineout, Montoya steered the maul over for the score and Henry’s conversion from the right made it 10-3 with 23 minutes played.
Warriors lost Weir to a shoulder injury on the half-hour and then were pushed back by a big hit from Lavanini in midfield and then Montoya’s work over ball on the floor.
The same two players then combined to put Freddie Steward over the tryline to the left of the posts, only for referee Andrew Jackson to blow for a forward pass.
When the scrum close to the Warriors line brought another whistle, Tigers at least had the consolation of another three points from Henry to stretch the lead out to 10 points befpre the break.
The restart was delayed while a breakdown in communication between referee and TMO were addressed
Both teams made changes to the front rows in the opening minutes of the second period, with Nephi Leaitgaga and Joe Heyes on for Tigers and both were involved as a lineout drive almost reached the tryline in the right.
And they both had a close-up view of a second score for Montoya from the base of another forward drive after Tgers old boy Graham Kitchener had been yellow-carded in defence of his own tryline.
Henry added the conversion to make it 20-3 with half an hour still to play.
After Henry picked out Van Wyk with a long pass on the right, Tigers pinned Warriors back on their line until Wiese beat the final tackle to score next to the posts for his first Premiership score and Henry’s extras made it 27-3.
Replacement prop Richard Palframan supplied am immediate response as he scored from close range for Worcester, and Jamie Shillcock kicked the conversion.
When Montoya left the field with a bloodied face, Tom Youngs emerged as a replacement in an historic appearance as he took his family total to 600 games in Leicester colours. And the club captain was straight on the ball in the middle of the field as Tigers looked to push on again.
Worcester wing Nick David sidestepped his way through broken field in the Tigers half but was outnumbered as he tried to set ball back and Van Wyk quickly latched on to the loose ball as referee Jackson whistled in Tigers’ favour.
Liebenberg’s intervention at the breakdown brought a penalty in a decent position for Tigers on the right with 15 minutes remaining on the clock and Henry took play into the right-hand corner where Youngs steer the charge over the line for his third try of the season and the bonus-point score.
Worcester answered with a score from Oli Morris as they trailed 34-17 with the game into the last 10 minutes.
A challenge on Liebenberg at a lineout gave Henry another kick into the right-hand corner and the maul ended with Wiese’s second score of the afternoon and Henry added an immaculate conversion.
Ted Hill provided the impetus for Worcester to threaten a third score and, although they were denied when the referee could not see any clear grounding as the back-rower claimed it, they retained possession for Perry Humphreys to dive into the right-hand corner and Shillcock added an excellent kick for the final points of the afternoon.