They were clinging on by their fingertips at the finish in Portlaoise but Meath are through to the Super 8s for the first time after edging an epic Round 4 qualifier.
Goals from top scorer Michael Newman and joint captain Bryan Menton proved decisive at the midlands venue as Meath bounced back from their Leinster final nightmare.
But it was anything but easy against a Clare side that played a fulsome role in a hugely entertaining contest that could have gone either way.
Meath led by four points entering the final quarter but were outscored by 0-6 to 0-3 in the closing minutes as Clare got the margin back to the minimum.
Time simply ran out on Colm Collins' team, however, as they came up just short of returning to the All-Ireland quarter-final stage having been there in 2016.
Clare's Championship is over but Meath will return to action next Sunday against Donegal in Ballybofey and will then play Mayo at Croke Park followed by a home game against Kerry in Navan.
It was a meeting of two well matched teams at O'Moore Park with the sides level on eight occasions in a rip-roaring first-half that finished 1-9 to 0-12.
Jamie Malone was terrific for Clare and helped himself to four first-half points, all from play.
Gary Brennan lined out at midfield but moved into the half-forward line soon after and powered through a tonne of good work.
He fisted over a ninth minute point but his moment of the game arrived close to half-time when he raced back up the field and dispossessed Cillian O'Sullivan who was bearing down on goal for Meath.
Malone's third point left Clare 0-10 to 0-9 clear when Menton struck Meath's opening goal in first-half stoppage time.
Padraic Harnan played in Donal Keogan - both of the defenders had huge games for Meath - and when his shot was cleared off the line by Kevin Harnett, Menton was on hand to drive in the rebound.
It should have left Meath sitting pretty at half-time but Clare, typically, came roaring back with points from Gordon Kelly and Malone to level it.
The second-half was just as exciting with Meath points from Newman and Conlon followed by a Gavin Cooney Clare goal.
Meath's response to the Cooney goal was emphatic and ultimately won them the game as Newman palmed in for their second goal.
He was injured in the process and was taken off but Meath still surged four points clear, 2-13 to 1-12.
It was a vital cushion that they relied upon as Clare rallied late on but the Royal County somehow held on.
Scorers for Meath: M Newman 1-6 (5 frees), B Menton 1-0, J Conlon 0-3, S Walsh (1 free), C O'Sullivan, B Brennan, B McMahon, T McGovern, D Keogan, E Devine 0-1 each.
Scorers for Clare: D Tubridy 0-6 (3 frees), J Malone 0-4, G Cooney 1-0, E Cleary (2 frees), C O'Connor 0-2 each, G Brennan, C O'Dea, G Kelly, D Coughlan 0-1 each.
MEATH: A Colgan; S Lavin, C McGill, S Gallagher; G McCoy, P Harnan, D Keogan; B Menton, S McEntee; E Devine, B McMahon, C O'Sullivan; J Conlon, M Newman, T McGovern.
Subs: T O'Reilly (blood) for Newman 43, G Reilly for McGovern 45, S Walsh for O'Reilly 54, S Curran for Gallagher 60, B Brennan for Devine 62, S Tobin for Conlon 62, O'Reilly for McMahon 75 (black card).
CLARE: S Ryan; K Harnett, C Brennan, G Kelly; D Ryan, A Fitzgerald, S Collins; G Brennan, S O'Donoghue; E Cleary, C O'Connor, J Malone; G Cooney, D Tubridy, C O'Dea.
Subs: K Malone for O'Donoghue 47, D Coughlan for Cooney 53, E O'Connor for Kelly 62 (black card), G O'Brien for D Ryan 64, E Courtney for Cleary 72.
Ref: D O'Mahoney (Tipperary).