Last year's All-Ireland finalists Down endured a tight squeak against Clare before hanging on to win by a point.
With Caolan Mooney making his debut in the qualifier, Conor Laverty's goal helped Down into a 1-2 to 0-0 lead.
However, Timmy Ryan's goal helped Clare draw level and Down led only 1-6 to 1-5 at half-time in the Ennis clash.
Down moved 1-11 to 1-6 ahead but Clare took control to level at 1-12 to 1-12 but Paul McComiskey's late point earned the Mournemen a fortunate win.
The Mournemen moved 1-2 to 0-0 ahead within five minutes as Laverty hit 1-1 and Martin Clarke also pointed.
Down could have been even further ahead by that stage as Laverty missed another excellent goal chance.
However, Clare were ahead by the 20th minute with full-forward Ryan scoring the goal and David Tubridy and Ger Quinlan also notching points.
Laverty brought his first-half tally to 1-4 as Down moved into a 1-6 to 1-5 advantage by the interval.
Down dominated the 10 minutes after the break with Mark Poland pointing after a Benny Coulter flicked effort had struck the crossbar.
Youngster Mooney was also on target as Down moved into a 1-11 to 1-6 advantage.
But Down's midfield problems began to seriously hurt them midway through the second half with Clare substitute Joe Dowling starting to lord possession at centrefield.
Two Dowling scores started a run of five unanswered Clare points which brought the sides level at 1-12 to 1-12.
These included a superb free from David Tubridy and an excellent John Hayes score.
The sides were still level as the game entered injury-time but a wrong option from Davy Russell allowed Down to win possession and they won a free after charging downfield.
McComiskey pointed the free to put Down ahead although the Mournemen had Laverty to thank in injury-time as he saved a goalbound effort on the line after Brendan McVeigh had failed to gather a long ball into the square.
However, Down held on to possession during the remainder of injury-time to edge the verdict.
Down: B McVeigh, A Brannigan, D Gordon, B McArdle, D Rooney, K McKernan, K Duffin, L Doyle, K King, C Mooney, M Clarke, M Poland, P McComiskey, B Coulter, C Laverty.
All-Ireland Football qualifiers first round
Clare 1-12 1-13 Down
Laois 2-16 0-11 Tipperary
Antrim 0-16 1-7 Westmeath
Louth 2-08 5-08 Meath
London 0-15 0-9 Fermanagh
Cavan 0-11 2-16 Longford
Wicklow 1-18 0-16 Sligo
Offaly 1-18 1-10 Monaghan