The hosts secured a 3-1 win in the five-match series with a comfortable triumph in the fifth Test at The Oval on Sunday, and Morgan, who was tipped as a possible replacement for Cook, has welcomed the change in fortunes.
"It's great to see Cookie do so well, particularly with the rough time that he has had," the Middlesex batsman told reporters. "It's fantastic to see both from a player's point of view and as a supporter.
"I don't see Cookie going anywhere anytime soon. I can't wait to be playing under him again, and I'm looking forward to the series.
"I think he plays in the ideal role to be captain – he leads from the front. Guys look up to him and he has tremendous respect around the international circuit, and more importantly in the changing room."
Morgan returns to action for England in the first one-day international against India next Monday.