England, who have not lost a home red-ball series since 2014, were 170-3 in 70 overs when a draw was agreed to leave this two-match contest all square ahead of next week's second Test at Edgbaston.
The Black Caps were in charge from the outset as opener Devon Conway marked his Test debut with a brilliant 200.
Southee (6/43) claimed six wickets spread over two days to dismiss hosts England for 275 in their first innings just before tea on Saturday.
This week's match will be the Blackcaps first fixture at the 'home of cricket' since their agonising loss Super Over loss to England in a 2019 World Cup final.
The Gloucestershire wicketkeeper-batsman is in line to make his Test bow as a wicketkeeper after fellow gloveman Ben Foakes was ruled out of the two-match series by a freak hamstring injury.
England have been reluctant recently to deploy James Anderson and Stuart Broad, their most successful all-time Test bowlers, in the same XI.
The 28-year-old sustained the injury after slipping in his socks while walking in the dressing room following Surrey's County Championship match against Middlesex at the Oval.