Luis Nani found substitute Danny Welbeck racing through on the left side of the Rangers box seconds earlier.
Welbeck, who had replaced Rodrigo Possebon only three minutes earlier, got the edge on his race for the ball with Peter Ramage and the Rangers defender brought down the United youngster.
Referee Phil Dowd pointed to the penalty spot and Tevez's looping run towards the ball ended with him coolly slotting home the penalty right-footed into the bottom left corner of the net, with Rangers goalkeeper Radek Cerny going the wrong way.
United found Rangers tough to break down with Cerny, in particular, in outstanding form.
He produced a string of magnificent saves both and after the goal as United's frustration grew against the solidity of the Rangers defence.
Cerny dived across goal to save Tevez's 25-yard shot early on and, although he spilled Anderson's 24th minute effort, he deserved his slice of luck.
United, with most of their top guns not even in the squad, moved up a gear after the interval.
In the 57th minute, Rafael's outrageous double turn on the right touchline took two Rangers defenders out of the game.
He touched the ball inside to Ji-Sung Park, whose rasping shot from the right side of the Rangers box, crashed against the inside of the right post.
Cerny denied Tevez with a brilliant save from his point-blank range effort soon after.
Rangers had a Samuel Di Carmine header disallowed for offside ten minutes later.
United ended with a flourish as both Park and Nani were denied by magnificent saves from the impressive Cerny.
But Rangers produced a sting in the tail and should have equalised deep into injury time - Emmanuel Jorge Ledesma shooting narrowly wide from 12 yards when well placed to leave United breathing a sigh of relief.