Mayo pushed on in the second half to register a 2-14 to 0-13 victory over Clare at Ennis.
With 11,576 in attendance, the losers put it up to their much-vaunted opponents in the opening period but Stephen Rochford's charges took a total stranglehold on proceedings after the restart, with brothers Cillian and Diarmuid O'Connor delivering the match-winning majors in the 42nd and 44th minutes respectively.
Victory means Mayo will be in fourth-round Qualifier action in a fortnight.
With the breeze at their backs, Colm Collins' plucky charges were full value for a two-point interval lead, 0-8 to 0-6, with last year's beaten All-Ireland finalists looking ruffled as they came under relentless pressure from a very impressive home team.
Andy Moran - a late addition to the starting XV - had the visitors ahead inside 15 seconds but the Banner County were quickly level from an Eoin Cleary free, with Donal Vaughan black-carded for a cynical pull-down on Gary Brennan.
David Tubridy swung over the Banner County's second point from a free off the ground in the ninth minute and Brennan brilliantly dropped over the point that made it 0-3 to 0-1 with ten minutes on the clock. A second Tubridy converted free increased the losers' lead and Clare were almost in for a goal on 15 minutes when Jamie Malone ran through the entire Mayo defence only to see his close-range effort saved by David Clarke when he had two free men inside for a certain three-pointer had he opted to lay the ball off.
Malone's excellent finish made it 0-5 to 0-1 at the midway stage in the first half as the hosts continued to dominate. When Keelan Sexton arrowed over a '45', the gap was five but back came Mayo with a Kevin McLoughlin point - their first score in 20 minutes - following an excellent Aidan O'Shea catch. Joe Hayes tipped a Diarmuid O'Connor blarger over the bar and the Connacht county were back within two point with twelve minutes left in the first half after Lee Keegan fisted over following a strong Moran run.
Clare centre forward Sean Collins increased the gap and the Banner were almost in for a goal again amid a frantic passage of play coming up to the half-hour mark only for a combination of inexperience and misfortune to thwart them. Sexton looped over the eighth Clare point on 33 minutes following superb approach play from the outstanding Martin McMahon but the favourites got the last two points before the short whistle courtesy of the excellent Moran.
McLoughlin gave Mayo a real boost when pointing within seconds of the restart and was denied by Hayes when the Clare net looked set to bulge. Mayo were ahead in the 42nd minute when a mistake from McMahon gifted them possession and Cillian O'Connor side-stepped Hayes before sticking the ball in the net - 1-7 to 0-8.
With Clare completely at sixes and sevens since the start of the second half, Diarmuid O'Connor took a pass from Cillian before drilling an unstoppable goal to the net to stretch the gap to five.
Tubridy's 48th-minute free represented Clare's first point of the second half but the winners replied with a classy point from veteran defender Keith Higgins. Cillian O'Connor's tap-over free had six between the teams at the three-quarters stage, 2-9 to 0-9.
O'Connor belted over another free from the ground 14 minutes from the end as the visitors continued to dominate and Clare's second point of the second half arrived from a Cleary free in the 59th minute. Cillian O'Connor (free) brought his tally to 1-3 and Man of the Match Seamus O'Shea's excellent solo point had the winners eight to the good with as many minutes left.
Substitute Gearoid O'Brien brilliantly supplied the losers' first second-half point from play in response to another O'Connor free and Cleary stroked over another late consolation. There were some unsavoury scenes in injury time after tempers became frayed following a converted Tubridy free.
A late, late O'Connor free provided the last nail in Clare's coffin as Mayo marched on.
Mayo - D Clarke; B Harrison, G Cafferkey, C Barrett; L Keegan (0-1), C Boyle, K Higgins (0-1); A O'Shea (0-1), S O'Shea; D O'Connor (1-1), D Vaughan, K McLoughlin (0-2); C Loftus, C O'Connor (1-5, 0-5f), A Moran (0-3). Subs: S Coen for D Vaughan (BC), E Regan for C Loftus, J Doherty for A Moran, T Parsons for S O'Shea, F Boland for D O'Connor, A Dillon for A O'Shea.
Clare - Joe Hayes; D Ryan, K Harnett, G Kelly; M McMahon, J Malone (0-1), C O'Dea; G Brennan (0-1), C O'Connor; P Lillis, E Cleary (0-3, 2f), C Russell; D Tubridy (0-4f), K Sexton (0-2, 1'45), S Collins (0-1). Subs: E Collins for C Russell (BC), John Hayes for C O'Connor, G O'Brien (0-1) for C O'Dea, S Malone for P Lillis, D Nagle for K Sexton (BC).
Referee - S Hurson.