Hawke's Bay's defence held firm and openside Josh Kaifa forced a penalty for holding on to allow them their first attack.
Tyrone Thompson threw to a five-metre lineout and got in the boot of a rolling maul to get the game's opening try.
Both teams continued to struggle for consistent attack, with the Magpies coughing up possession, but disrupting Otago's set-piece play on defence.
Otago's territorial advantage paid off in the 29th minute when number eight Rupeni Tamani won a penalty for holding on and Millar kicked their first three points.
Hawke's Bay undid much of their good defence soon after when miscommunication between Tuipulotu and prop Mark Braidwood left a big hole in the line for Josh Ioane to score an easy try.
The Magpies rode another lineout drive into Otago's red zone before the hosts were penalised for offside a couple of minutes before halftime.
Hawke's Bay were happy to take three points through fullback Harry Godfrey's boot and go into halftime tied at 10-all.
The deadlock lasted less than five minutes before a Millar 50/22 kick gave Otago a five-metre lineout from which hooker Ricky Jackson scored a try.
Hawke's Bay withstood more attacking pressure then built back into the game through consecutive penalties towards the hour mark.
Handling errors continued to cost them and Millar missed a shot from halfway after the Magpies were caught offside in the 64th minute.
The first five-eighths made no mistake from 25 metres out shortly after with his penalty goal, giving Otago a two-score lead.
Godfrey broke past the 22 metre line and Hawke's Bay had some good attacking phases before yet another knock-on with 10 minutes remaining.
The Magpies kept the pressure on and Otago lock Josh Hill received a yellow card for a cynical penalty on defence.
Lolani Faleiva had a try scrubbed off for a knock-on at the base of a preceding ruck with five minutes to play.
Godfrey slotted home a penalty goal with the penultimate play and Hawke's Bay made it over halfway after the hooter but lost the ball and had to settle for a losing bonus point.
The Magpies will hope to make amends against winless Manawatū in Palmerston North on Sunday.
Otago 18 (Ioane, Jackson tries; Millar 2/4 pens, 1/2 cons)
Hawke's Bay 13 (Thompson try; Godfrey 1/1 con, 2/2 pens)
HT: 10-10
