Rami Al Hajj marked his first Argyle start with a goal after eight minutes.
Substitute Ibrahim Cissoko then scored twice in the second half, either side of Victor Moses's reply for the visitors.
"I am delighted for the lads," said Rooney. "It was a good performance for different reasons tonight, in the first half we controlled the tempo of the game.
"Luton tried to press us and when we played with composure and through the press we caused them problems.
"Second half we were a little sloppy and gave the ball away a little bit too often but our back four were excellent.
"What you are seeing is a real togetherness, everyone at the club - staff and the fans - everyone is buying into what we are trying to do. There's a really good feeling around the club."
Luton boss Rob Edwards felt his side were outplayed in key areas of the pitch.
"Credit to the opposition. Both ends of the pitch they were better; they were more ruthless at one end and defended better at the other," said Edwards.
"I think we got into their final third 40 odd times, twice as often as they did but we didn't make their goalkeeper work enough.
"We weren't clinical enough at one end and then we didn't defend well enough at the other and that's not a good recipe to win a football match.
"I am convinced we are going in the right direction and things will go our way soon."
Plymouth moved to 13th in the table after notching only their second win of the campaign while Luton, who were relegated from the Premier League, are 16th in the 24-team division.
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