Southee, Somerville and Ajaz took three wickets apiece to earn Kiwis a fantastic 123-run victory over Pakistan to take the series 2-1.
Kane Williamson scores unbeaten ton puts visitors in command after Yasir Shah creates history.
He broke 82-year-old record, Australian Grimmett set the previous record in 1936 when he took 200 wickets in 36 matches.
Azhar Ali and Asad Shafiq slam centuries to give hosts upper-hand on third day of final Test.
Pakistan batsman Mohammad Hafeez has announced that he will retire from Test cricket at the end of the ongoing Test against New Zealand in Abu Dhabi.
The hosts were 139/3 at stumps on second day, 135 runs adrift of the Black Caps total.
Yasir Shah and Bilal Asif put hosts in control despite Williamson resistance, visitors ended the day on 229 for 7.
The leg spinner picked 14 wickets in the match to help Pakistan win by an innings and 16 runs.
Yasir Shah enjoyed a day straight out of a dream, picking up ten wickets, as Pakistan enforced a follow-on upon the visitors on day 3 at Dubai.
In response, New Zealand were steady, as their openers took to them safely to stumps.