The opening goal was a well-executed free-kick after 25 minutes, while the second was a free header at the back post after 54 minutes.
The struggling hosts were poor in most areas of the Moses Mabhida Stadium pitch and were comfortably beaten by Roger de Sa’s men, who moved to 34 points, one behind log-leaders Kaizer Chiefs.
Abafana Bes’thende, meanwhile, dropped to 13th after a seventh loss of the season.
Klate had the first opening after seven minutes when Andile Jali’s ambitious cross-field ball somehow reached the winger, cutting in from the left, but the final attempt was drilled over.
Arrows’ first opportunity was nothing more than a half-chance as Bongi Ntuli collected a long pass from midfield, got the better of Lucky Lekgwathi and dragged his 20-metre shot across the face of goal.
Klate was again in involved midway through the half when his clever free-kick to the edge of the box caught out the home defence, but Oupa Manyisa pushed the attempt wide.
At the other end, Ntuli had the best opening of the game when Siyanda Zwane beat his man on the left and rolled across the face of goal, with his teammate somehow blazing over.
But there was no let-off for Arrows the next time the Buccaneers had a chance, a long-range free-kick on the right. Klate stepped up and bent his set-piece into the right corner.
Collins Mbesuma and Thabo Matlaba had two further opportunities either side of the half-hour mark as Pirates continued to dominate against a poor home team.
It remained that way until the break and after the restart the visitors maintained the tempo, with Manqoba Mngqithi’s men trying to keep a foothold in the game.
The hosts had a chance to pull level moments later, but Musa Bilankulu wastefully sent his header over from a flighted Joseph Musonda corner.
Mbesuma should have made it 3-0 against his former club, but goalkeeper Pavels Steinbors collected the ball at his feet after the Zambian was through one-on-one.
With time running out, Bilankulu had a header cleared off the line as the defending champions held on for an easy win.