Puel (above) also left out Vardy and made changes to his starting line-up when Leicester were knocked out of the Carabao Cup in the quarter-finals, beaten in a penalty shoot-out by Manchester City.
City added to their lead in the second half with Gabriel Jesus tapping in from Sterling's cross and Riyad Mahrez firing in after playing a one-two with Ilkay Gundogan.
"It is one you have to savour", Oldham boss Pete Wild said.
Morgan, though, believes the minnows' win was undeserved, explaining: "It was a hard day and the magic of the FA Cup happened, definitely".
Denis Odoi put Fulham ahead in the 52nd minute at Craven Cottage, but Sam Sturridge drew Oldham level with a penalty before Callum Lang's header two minutes from fulltime sealed a comeback victory.
Sheffield United began the day 84 places above Barnet but were outplayed by their lowly opponents for whom former Tottenham trainee Shaquile Coulthirst won it with a 21st-minute penalty after Richard Stearman brought down Ephron Mason-Clark.
His Cardinals didn't let him down, hanging with the Hornets aside from Will Hughes' 13th minute goal and Troy Deeney's late insurance marker. We had loads of half-chances, some very good positions where we should score, and on another day they all go in.
"I think we lost together, we are all together disappointed and frustrated after this game, because it was an objective for us".
The 7-0 defeat to Man City was Rotherham's heaviest ever defeat in the FA Cup.
Elsewhere, in an all-Premier League clash, Brighton won 3-1 away to south coast rivals Bournemouth.
Premier League sides come into play in the third round proper (round of 64), with lower league teams battling it out in the early stages of the competition.
Premier League leaders Liverpool will complete a third round that started with Tottenham´s 7-0 thrashing of Tranmere on Friday when they travel to top-flight rivals Wolves on Monday.