Aguero, however, levelled from a quickly-taken free-kick and made it 2-1 from Ilkay Gundogan's back-heel.
Sub Bernardo Silva added a fourth after Sane had beaten Pope to a threaded pass from sub Kevin de Bruyne and squared the ball for an easy finish.
"Both of Sergio Aguero's goals were awesome, there were clever and quick and I am happy with both of his goals", Guardiola told reporters.
"It's an honour for me to have Aguero under my command".
The City chief said: "I apologise, I didn't want to go out of control".
"The quality he has inside the box is fantastic and both of his goals were awesome".
"We play the first leg here at home, so, of course, we want to win that game". The Argentinian, Guardiola said later, was crucial to City's success and would stay at the club.
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Yet Lys Mousset reduced the deficit for Bournemouth in the 55th minute and League One leaders Wigan, who won the FA Cup in 2013, were denied a surprise win by Steve Cook's goal two minutes into stoppage-time.
Guardiola made headlines earlier in the week when he suggested the heaviness of the festive fixture schedule was "going to kill" players. "Against Huddersfield (last season), we needed a replay".
It will be interesting to see if Guardiola again risks his Player of the Year candidate or if he gets a rest 12-day rest until their next Premier League match against Liverpool on Sunday. Now he's going back to Spain and he'll be back when he feels good, we are happy because everything is going well.
"David is a legend for the club and for what he has done", said Pep.
Manchester City have led by more than 1.5 goals at halftime in just 1 of their last 7 games in all competitions.
Dyche shrugged off the incident, saying: "You both fight, you both want your teams to win". "That's how it should be".
"Fortunately it was allowed and I think it was also correct".
"The referee said he didn't have to blow the whistle, but Ashley Westwood was five yards away and the ball went through the gap where he should have been". "I'm bound to be aggrieved by that".