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Casemiro missed a spot kick for United in the shootout but they hit back and Rasmus Hojlund's penalty sent them into the final against Manchester City.
Silva sidefooted the ball home in the 84th minute after Chelsea goalkeeper Djordje Petrovic parried a shot from former Blues midfielder Kevin De Bruyne.
With seven games remaining in the neck-and-neck title race, Liverpool and Arsenal have 71 points with the Gunners leading on goal difference. Holders Manchester City are third on 70 points, while United are sixth.
Juergen Klopp's side have 70 points from 30 games, two ahead of Arsenal and three more than champions Manchester City.
Manchester United fanned the embers of their top-four hopes with a 2-0 win against visiting Everton.
Erling Haaland, who missed a golden first-half chance, secured the points with a late strike.
A point moves Tottenham level with north London rivals Arsenal, just outside the top four only on goal diference.
Manchester City slipped out of the top four as Tottenham Hotspur made up ground with a third consecutive victory, beating Everton 2-1 at home.
Diogo Dalot was sent off in stoppage time for two yellow cards -- both for dissent.
Three-time Champions League winner United needed to beat Bayern Munich at home and hope for a favorable result in the other Group A game but lost 1-0 to the six-time winner.