The table-topping Lilywhites held a 0-4 to 0-3 lead after a defensive first half in a rain-lashed Pairc Esler.
Slattery netted just 90 seconds after coming on as a half-time substitute and he fired in a second goal to give Kildare a six-point cushion.
Down stay fifth with two wins and three defeats.
There was little to entertain the Newry crowd in the first half although Down went close to an early goal when Ryan Johnston's shot was saved by Mark Donnellan.
The sides exchanged the opening points with Barry O'Hagan firing over two for the hosts while Donnellan and Tommy Moolick were the Kildare scorers.
Kevin Feely's double gave Kildare a 0-4 to 0-2 lead before Darragh O'Hanlon halved the gap before the half-time whistle.
Feely tagged on the first point of the second half and he set up Slattery to hit the Down net after a poor kick-out.
Caolan Mooney replied with two Down points and Kevin McKernan added another after Feely registered his fourth score of the evening.
Slattery banged in his second goal past keeper Michael Cunningham to leave Kildare 2-6 to 1-6 in front.
The game fizzled out as a contest and Down finished with 13 men after black cards for Alan Davidson and Jerome Johnston when all their susbstitutes had been used.
Allianz Football League
Saturday - 19:00 GMT
Division One
Donegal 0-12 0-6 Tyrone, Ballybofey,
Kerry 0-13 0-13 Dublin, Austin Stack Park
Division Two
Down 0-10 2-9 Kildare, Pairc Esler
Division Four
Carlow 2-10 0-7 Waterford, Dr Cullen Park
Down were undone by David Slattery's two second-half goals as the Mourne side fell to a 2-9 to 0-10 defeat by Kildare in Division Two.