Despite being outclassed for the majority of the match, home fans had plenty to cheer about in the final quarter of the match as Mark Bright (2), Andrew Goodman, Tom Marshall and Taula Koloamatangi all touched down for tries.
The win sees the Cantabs rise to second behind Bay of Plenty at the top the Premiership table, but Tasman remain at the foot of the Championship on six points with North Harbour.
In the first stanza, Osbourne (2) and Veainu scored to lead 19-3 at the break.
Tasman's points came via the boot of captain Andrew Goodman with a penalty.
Osbourne opened the scoring for the red and blacks in the 3rd minute of the match, with fullback Sean Maitland getting on the outside of his marker to set his winger away to score a try.
First-five Tyler Bleyendaal kicked the extras.
Opposite wing Veainu made a a difficult try look easy as he beat three players down a narrow sideline corridor to squeeze the ball inside the corner post to score in the 25th minute. Bleyendaal converted.
Fijian winger Osbourne claimed his second try, as he caught Bleyendaal's cross-kick before rounding the fullback on his way to the tryline.
Canterbury 36 (Patrick Osborne 2, Telusa Veainu 3 tries; Tyler Bleyendaal 2 con, Tom Taylor pen, 2 con)
Tasman 32 (Mark Bright 2, Andrew Goodman, Tevita Koloamatangi, Tom Marshall, tries; Goodman pen, 2 con). Halftime: 19-3.
- HERALD ONLINE