"Even though we gave up a couple of soft ones, it still felt like we were in control out there for the most part," said Livesey, who is back from a season-ending knee injury in 2011.
"We definitely deserved the three points after that first half performance but we need to learn to close out wins from 3-0 with a lot more composure.
"There might have been a bit of fitness [lacking] but we sat back and let them play in the second half and didn't show a lot of nerve."
Tauranga took a quarter to settle into their work on the hard Mt Maunganui surface that made for an awkward passing game.
But once they found some rhythm, they cracked the Onehunga-Mangere defence, with Livesey first to score when he leapt high and met a Ben Van Der Salm cross to the far post with his head, cleverly looping the ball back across goal into the open net in the 23rd minute.
As the chances started to mount it was inevitable Tauranga would double their lead, with Livesey getting his second from the penalty spot just minutes before halftime after striker Scott Doney was brought down in the penalty area by James Pearce.
Jindrich Hahn, who missed Saturday's slaughter and started off the bench, added a third midway through the second half with his trademark control.
The Czech national turned his marker, danced past three more defenders on his way into the penalty area before sliding a left-footed shot into the bottom corner.
But the wobbles came, with woeful defending letting Paul Sluyter score direct from a swinging corner kick, before Onehunga-Mangere skipper Parusi Lemalu sliced the lead further with a speculative effort from 25m that caught the defence flat-footed.
Suddenly it was game-on, with Onehunga forcing a wave of corners, although Tauranga midfielder Jordan Culpepper had a couple of chances of his own to seal the win but fired wide.
The jury's still out on former UK pro Ian Stringfellow's effectiveness playing a wide role on the right but there was enough there yesterday to suggest City have the squad to push clear of the Northern league's lowest rung.
Livesey is just pleased to be back after spending nearly all of last season sidelined. "The knee's holding up well but the rest of the body's feeling it a bit," the 33-year-old said. "This hard ground's not too good for us oldies but getting on the scoresheet should help ease the sore muscles."
Tauranga's next game is on Saturday at Western Springs, with a return home on April 22 against North Shore.
Northern League second division scores:
Yesterday:
Tauranga City United 3 (Tom Livesey 2, Jindrich Hahn) Onehunga-Mangere 2 (Paul Sluyter, Parusi Lemalu). HT: 2-0
North Shore Utd 3 (Jaryd Gray 2, Chris Bryson) Mangere Utd 0. HT: 1-0.
Waiuku AFC 0 Warkworth 3 (Steve Gove 2, Rob Carty). HT: 0-2.
Matamata Swifts 1 (Mark Knell) Fencibles Utd 1 (Stephen Lawry). HT: 1-0.
Western Springs 1 (Luke Petrie) Hibiscus Coast 0. HT: 1-0.
Manukau City 2 (Alvin Kumar 2) Papakura City 2 (Chris Abbott 2). HT: 1-2.
Saturday:
Hibiscus Coast 4 (Trent Mayes 3, Daniel Cyra) Tauranga City 1 (Graham Craven). HT: 1-0.
North Shore United 1 (Jaryd Gray) Waiuku 1 (Daniel Baldwin). HT: 1-0.
Onehunga-Mangere 1 (Parusi Lemalu) Mangere United 1 (own goal). HT: 1-1.
Papakura City 1 (Nial Okakpu) Western Springs 2 (Luke Petrie, Dean Brereton). HT: 1-0.
Fencibles 0 Manukau City 2 (Michael Day, William McClure). HT: 0-2.
Warkworth 0 Matamata Swifts 1 (Stu Watene). HT: 0-0.