Slater made a stylish 94 to guide the Sheffield Shield champions to their third successive win against England.
Michael Atherton's side have just one four-day match, against an Australian XI in Hobart, to lift themselves before the first test in Brisbane starting Nov. 25.
England, comprehensively outplayed with both bat and ball, spent much of the day facing the prospect of a much heavier defeat.
Beginning the day on 200 for five, a lead of 163, the tourists lost their last five wickets for just 44 runs as left-arm spinner Anthony Kershler and pace bowler Wayne Holdsworth finished with three wickets each.
The home side, set a target of 208, appeared to be cruising to victory as they reached 162 for two. Slater took the opportunity to have a good look at England's attack.
But Slater's dismissal, caught and bowled by left-arm spinner Philip Tufnell after facing 140 balls and hitting 10 fours, prompted a brief English revival.
New South Wales slumped to 184 for six before Neil Maxwell, who made 21 and wicketkeeper Craig Glassock, with 15, secured victory with an unbroken seventh-wicket partnership of 27.
England's cause was not helped during the morning session when former captain Mike Gatting and spinner Shaun Udal fell to reckless shots.
Gatting failed to add to his overnight 33 before being stumped, while Udal carelessly drove the second delivery with the second new ball straight to mid-off with his score on nine. It was left to all-rounder Craig White, with a fighting 46 including nine fours, to take England's lead past 200.
Opening the New South Wales innings, Slater and his state and test captain Mark Taylor, who made 47 to add to his first innings century, attacked some disappointing bowling to put on 94 in only 22 overs.
Atherton denied any panic in the England camp, although he admitted his side's performance was unsatisfactory.
"I'm not overly concerned about things but I thought that in all areas we were slightly sub-standard," he told reporters. "I'd prefer to win every game but there certainly isn't a crisis."
While the England captain clearly has work to do, his Australian counterpart is clearly relishing the prospect of facing the tourists again.
"I'm definitely confident for Brisbane after this match and if we can play to our potential we should be good enough to retain the Ashes," Taylor said.
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