Mayo recovered from a poor opening half to kill off Clare with two goals in a three-minute spell just after half-time.
The O’Connor brothers, Diarmuid and Cillian, delivered the two goals which set Mayo on their way and broke Clare’s hearts in a great contest.
The hosts gave themselves every chance of victory when they led by 0-08 to 0-06 at the interval having played against the strong breeze in front of 11,576 at Cusack Park.
But it was the All-Ireland hunters Mayo who dominated after raising the green flags.
It took Mayo just 15 seconds to get off the mark when Andy Moran, one of two late changes, pointed but they suffered a big blow two minutes later when Donal Vaughan picked up a black card for pulling down Gary Brennan.
Brennan was superb for Clare in the opening half and with Martin McMahon a constant threat running from deep, the Banner men got on top.
Eoin Cleary levelled with a point and then they took over with David Tubridy landing two frees either side of an effort from the impressive Brennan to lead by 0-04 to 0-01 after 14 minutes.
Jamie Malone soloed through from deep but his goal effort was superbly saved by David Clarke, on a day when the Mayo custodian was making his 100th appearance for the county.
Clare extended their lead when Malone pointed from the right and then Keelan Sexton landed a ’45 to lead by 0-06 to 0-01 after 20 minutes.
Aidan O’Shea helped Mayo respond, setting up Kevin McLoughlin for a good point and this was followed by efforts from Diarmuid O’Connor and Lee Keegan.
But Clare again responded in style with Sean Collins finishing off a good move with a point and Sexton made it 0-08 to 0-04 approaching the break with another fine shot.
Mayo finished the half strongly with veteran Moran firing over two points to leave it 0-08 to 0-06 at the break.
Mayo took control after the restart, with Kevin McLoughlin pointing just 21 seconds after the restart before they blasted two goals in a three minute spell from the O’Connor brothers.
Conor Loftus set up the first one after 42 minutes when he intercepted a free from goalkeeper Joe Hayes before sending Cillian O’Connor through to shoot to the net.
They pushed on and a good passing movement started by corner-back Chris Barrett was blasted to the net by Diarmuid O’Connor to make it 2-07 to 0-08.
Tubridy pulled back a free for Clare but they struggled to cut the gap as Mayo controlled most areas of the field and Cillian O’Connor kept putting further distance between them as Mayo advanced to the next round.
Gearoid O’Brien got Clare’s first score from play in the second-half with just four minutes remaining, with Eoin Cleary then landing one from the left wing.
But by then Mayo were out of sight after a dominant second-half display.
Clare: Joe Hayes; D Ryan, K Hartnett, G Kelly; M McMahon, J Malone (0-01), C O’Dea; G Brennan (0-01), C O’Connor; P Lillis, E Cleary (0-03, 0-02 frees), C Russell; D Tubridy (0-04, 0-04 frees), K Sexton (0-02, 0-01 ’45), S Collins (0-01).
Subs: E Collins for Russell (41 mins, black card), John Hayes for O’Connor (41 mins), G O’Brien (0-01) for O’Dea (52 mins), S Malone for Lillis (65 mins), D Nagle for Sexton (76 mins, black card).
Mayo: D Clarke; B Harrison, G Cafferkey, C Barrett; L Keegan (0-01), C Boyle, K Higgins (0-01); A O’Shea (0-01), S O’Shea; D O’Connor (1-01), D Vaughan, K McLoughlin (0-02); C Loftus, C O’Connor (1-05, 0-05 frees), A Moran (0-03).
Subs: S Coen for Vaughan (4 mins, black card), E Regan for Loftus (48 mins), J Doherty for Moran (55 mins), T Parsons for S O’Shea (58 mins), F Boland for D O’Connor (65 mins), A Dillon for A O’Shea (67 mins).
Referee: Sean Hurson (Tyrone)