For Guardiola, this was a signal that his City team can fashion wins against the most obdurate opposition.
Pope dived to his left to save well from Kevin De Bruyne, who had been played through by David Silva's sublime pass, but as the ball ran loose, the goalkeeper was judged to have clipped Bernardo Silva, who fell to the ground. Burnley fumed, convinced of a dive.
The successful penalty was Aguero's 177th Manchester City goal since he joined the club from Atletico Madrid in 2011.
Burnley put up a stubborn effort and made it hard for City to break them down. Sadly for Wood, he was injured in the move, and had to come off minutes later.
Aguero, back in the starting lineup after breaking a rib in a auto crash three weeks ago, converted a penalty kick in the 30th minute at Etihad Stadium. Bernardo Silva should have opened the scoring after Sane skipped around a challenge and pulled the ball back but the Portuguese shot straight at Nick Pope.More news: Manchester United expect Zlatan Ibrahimovic to return before end of year
He has scored some sensational goals during the intervening six years, most memorably winning the Premier League title with a dramatic strike in City's 3-2 victory over QPR on the final day of the 2011/12 season. The German raced into the area and lashed a low shot past Pope.
City had to survive a couple of awkward moments early in the second half, with their opponents claiming a penalty as Fernandinho accidentally volleyed a clearance on to his own hand, although East was right to reject the appeals.
City found it tough to create clear chances in the second half as Burnley continued to be disciplined, with Pope blocking David Silva's shot at the near post after good approach play from Aguero and Sane.
Bernardo insisted that he was caught by Pope for the penalty in the 30th minute. However, in the space of a couple of minutes, City scored two quick goals that put the match out of their opponents' reach.
Otamendi rose highest from Sane's corner to power a header into the corner, despite the presence of Steven Defour on the post.
Moments later the points were wrapped up as De Bruyne picked out Sane with a characteristic defense-splitting through-ball.