The game remained on a knife's edge for most of the second half, and England couldn't convert their rare raids on the Australian line into points.
In front of a crowd of 22,724, England utility James Roby almost reduced the lead when he burrowed his way from dummy-half but was held up over the line.
Slater, who finished with a game-high 200 metres in his first Test since 2014, also helped save another possible England try late in the game by pushing Ryan Hall into touch. Cameron Smith kicked the advantage to eight points with a penalty goal in the 76th minute, before Josh Dugan sealed the win with an 80-metre runaway try in the closing minutes.
An upset appeared on the cards when England drew first blood through Gareth Widdop's dubious long ball for winger Jermaine McGillvary in just the fourth minute.
The Kangaroos had a number of opportunities to reply, including three-straight sets on the England line, but were rusty in attack and were turned away each time.
It wasn't until Matt Gillett stepped off his right foot and found space in the 23rd minute that the home side equalised, before taking the lead with Slater's try not long after.
The veteran fullback dummied his way past a tiring defence in the 29th minute, giving Australia a four-point lead at the break.
McGillvary was easily England's best by almost matching Slater's 200-metre output.
As it happened:
