Brazilian Piquet was asked to deliberately crash to help team-mate Fernando Alonso to win the race.
Among those to congratulate Hamilton was former McLaren team-mate and double world champion Fernando Alonso.
Spain's Fernando Alonso has been tipped as a possible replacement for Yoong in Hungary and Belgium.
Vettel beat Ferrari's Fernando Alonso by 0.076 seconds, with the Spaniard's team-mate Felipe Massa in third.
Robert Kubica took second for Renault, ahead of the Ferraris of Felipe Massa and Fernando Alonso.
Mark Webber, Fernando Alonso, and Jenson Button will start in second, third and fourth position.
And Fernando Alonso got big cheers on the basis that he clearly doesn't like Hamilton.
Massa's teammate and world championship leader by 24 points, Fernando Alonso will start down in 10th.
Kimi Raikkonen lines up third for Ferrari alongside Renault's Fernando Alonso, who did well to qualify fourth.
Renault's Fernando Alonso was fourth, unable to stay with the Ferraris, but well clear of everyone else.
In 2012, for instance, Vettel averaged 1.85 stops per race and Ferrari's Fernando Alonso was on 1.75.
Fernando Alonso is said to be keen on having the 2009 world champion as a future team-mate.
BBC: A photograph of the main grandstand at the Indian GP circuit
Renault's Fernando Alonso was left disappointed again after finishing eighth despite starting third on the grid.
Ferrari's Fernando Alonso starts in third ahead of the McLarens of Jenson Button and Lewis Hamilton .
Should he return, Schumacher would go head-to-head with fellow world champions Jenson Button, Lewis Hamilton and Fernando Alonso.
The German's best time of one minute 20.087 seconds was almost half a second faster than Fernando Alonso.
BBC: Red Bull leave rivals trailing in Hungarian GP practice
Schumacher is now just two points behind world champion Fernando Alonso in the race for the drivers' title.
BBC: SPORT | Motorsport | Formula One | Schumi 'had no pressure to quit'
And he was third fastest overall on Saturday, behind Nick Heidfeld in the Renault and Fernando Alonso's Ferrari.
Ferrari's Fernando Alonso was the long-time pace-setter but Heidfeld beat him by 0.132 seconds late in the afternoon.
BBC: Nick Heidfeld heads Fernando Alonso at his Renault test
McLaren's Jenson Button starts on the second row, ahead of Red Bull's Mark Webber with Ferrari's Fernando Alonso.
BBC: Sebastian Vettel beats Lewis Hamilton to Abu Dhabi pole
Australian GP runner-up Robert Kubica finished sixth in his Renault, ahead of current world championship leader Fernando Alonso.
Championship leader Lewis Hamilton and McLaren team-mate Fernando Alonso have apparently called a truce following their very public spat.
BBC: SPORT | Motorsport | Formula One | Turkish GP qualifying as it happened
Schumacher will be looking to cut further into Fernando Alonso's world championship lead after claiming his 68th career pole.
Juan Pablo Montoya was a distant third for Williams, ahead of the Renaults of Fernando Alonso and Jarno Trulli.
BBC: SPORT | Motorsport | Formula One | Schumacher in total control
The German also apologised to Fernando Alonso but again said he had not halted his last qualifying lap intentionally.
BBC: Schumacher and Todt faced a barrage of criticism in Monaco
Ferrari's Fernando Alonso was quickest in both Thursday practice sessions, but the Spaniard expects a hard fight in qualifying.
Vettel carries a 13-point lead into the race over Ferrari's Fernando Alonso of Spain, who placed third in Texas.
WSJ: Formula One: Lewis Hamilton Wins U.S. Grand Prix in Texas
But in a damp second practice Hamilton trailed in ninth, well behind Massa who was second to Fernando Alonso.
Vettel arrived at the Yas Marina Circuit 15 points behind then-leader Fernando Alonso of Ferrari and seven adrift of team-mate Mark Webber.
BBC: Red Bull's Sebastian Vettel kept Formula 1 title belief
Ferrari's Fernando Alonso won on home soil, with Red Bull's Sebastian Vettel and Mark Webber finishing fourth and fifth respectively.
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