The company has been fined $750,000 for its errors, including $150,000 for each of two escapes from custody.
One of the inmates, Graham Hay, escaped after the handcuffs that were placed on him for a medical trip were too loose-fitting.
Serco was being closely scrutinised because it has been given the contract for a new 960-bed prison in Wiri, to open in 2015.
Ms Stevenson noted that the company was showing signs of improvement. Its report card showed it had drastically reduced drug use in the Mt Eden facility, and it had passed 34 of its 37 targets last month.
Reducing drug use was one of the key performance indicators in its $300 million contract, and the company was expected to ensure less than 12 per cent of inmates tested positive for drugs.
After 12 months, 3.88 per cent of prisoners had been found with drugs in their system.
The Corrections Department did not collect performance measures for public prisons by individual facility, and therefore it was difficult to compare their record with Serco's.
But its annual report showed taxpayer-funded prisons had a positive drug test rate of 7 per cent.
Prison reform lobbyist Kim Workman said Serco's drug programme could only be called successful if prisoners stopped using drugs when they left jail.
It was crucial that drug addiction - not just drug use - was reduced.
Serco's first year
* Rate of positive drug tests: 3.88 per cent (target: maximum of 12 per cent)
* Rate of self-harm: 0.11 per 100 prisoners (target: less than or equal to 0.7)
* Wrongful detention of prisoners: 3 (target: 0)
* Deaths in custody: 0 (target: 0)
* Prisoners with a management plan within 28 days: 42 per cent (target: 90 per cent).