Wagner takes 6 for 41 to restrict the hosts to 164 on Day 1 of the first Test at Bulawayo on Thursday. NZ were 32-0 at stump.
The Indian cricketer and his actress wife welcomed the little one with their families in London.
Kusal Mendis shared 117 runs for the fifth wicket with Dinesh Chandimal (42) after Sri Lanka had slumped to 86 for four.
The visitors earned a handy lead of 86 runs before Mitchell Starc removed Perera on another truncated day.
He replace Yasir Shah after returns of 7 for 83 in the second innings of the Antigua. Root moves up to No. 2 among batsmen
Josh Hazlewood and Nathan Lyon shared six wickets bowled out Sri Lanka for 117. Australia were 66-2 at stump.
England thrashed Pakistan by 330 runs in the second Test at Old Trafford to level the four-match series 1-1.
After a gap of four years since their previous tour in 2011, Australia are set to play three Tests on the Sri Lankan soil, starting Tuesday.
Zimbabwe strike bowler Tendai Chatara has been ruled out of the test series against New Zealand after hurting his ankle, Zimbabwe Cricket announced on Monday.
At stumps, England looking to level the four Test series at 1-1, were 98 for one with Alastair Cook 49* and Joe Root 23* in the middle.