Paceman Hazlewood led the way with three scalps on the second morning as the Australians bounced back after Wednesday’s horror day in the field in sweltering conditions.
Haddin equalled the Australian record of six dismissals in an innings held by Wally Grout, Ian Healy and Rod Marsh with two more catches on the second morning.
Lunch was taken upon the final wicket with Umesh Yadav caught in the deep by Chris Rogers off Nathan Lyon for nine.
Varun Aaron had been caught by fielding substitute Marnus Labuschagne off Lyon for four on the last ball before lunch forcing extended playing time for the ninth wicket down.
The Australians began the day well with two wickets in the opening 40 minutes before captain Mahendra Singh Dhoni and Ravichandran Ashwin regained the initiative with a belligerent seventh-wicket stand.
The pair put on 57 in equal time before Ashwin became Hazlewood’s fourth wicket, playing away from his body and edging chest-high to Shane Watson at slip for 35.
Dhoni cracked 33 off 53 balls with four fours before he fell to the Hazlewood-Haddin combination.
Hazlewood was far and away Australia’s best bowler with five for 68, while left-armer Mitchell Starc continued to struggle with his line and length and was punished, going wicketless for 83 runs.
Hazlewood struck early enticing an edge off Ajinkya Rahane with a late outswinger to give Haddin his fifth catch of the innings.
Rahane hit 81 off 132 balls with nine fours and put on 60 for the fifth wicket with Rohit Sharma.
Steve Smith pulled off a magnificent sprawling catch at second slip to dismiss Sharma for 32 off Watson in the 92nd over.
Smith anticipated the catch, flinging himself to his right to pluck the ball just off the ground to the delight of his teammates.
Starc returned to bowling after Test great Shane Warne called his body language “soft” during commentary for Network Nine on Wednesday.
He got a couple of deliveries to go through to the keeper but was hit for successive fours by Dhoni and again proved ineffective with his wayward line and length.