It was their sixth successive qualifying Group Six win under Fabio Capello, who Terry credits for much of the transformation after ex-manager Steve McClaren's failure to qualify for Euro 2008.
"The players are now really relishing getting on the ball and playing and having confidence and trusting each other again," added Terry.
"There were times when people were on our backs a little bit and certain players didn't want the ball in certain areas.
"But if you look at everyone in this game, they all wanted the ball and were all making angles for each other.
"Since the manager has come in, that has been a big part of it, to play right from the back and work hard for the team. You can see how much harder we are working as a group."
Barry, who completed a £12m move to Manchester City on Tuesday, broke the deadlock on 40 minutes before former Aston Villa team-mate Heskey tapped home a second just before half-time.
Rooney made it 3-0 with a spectacular strike before Lampard completed the scoring with a well-taken penalty.
"I'm happy with the team's performance," said England boss Capello. "Kazakhstan started strongly but step by step we improved.
"When you play a team that's not at the same level you have to score. After scoring the first, it was made easier."
Barry, who will miss Wednesday's game with Andorra after getting booked against Kazakhstan, said: "We got everything we expected from Kazakhstan.
"They had a lot of possession and created a couple of chances and we had a couple of scary moments.
"We knew the first goal would be massive and it took the steam out of them. I was delighted to score the opener that made everyone settle - it was an important stage of the game."
Capello's side can make it seven out of seven when they face bottom-placed Andorra at Wembley on Wednesday.
When asked about Barry's suspension, Capello joked: "After this performance I think he can go on holiday. He scored a goal, he signed a new contract and he changed his team. It's been a fantastic weekend for him."
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