However, the All-Ireland junior champions of two seasons ago had enough in hand to see them through.
David O’Neill put Mayfield two points to one ahead after 11 minutes, and they made the first decisive break when Shane Duggan goaled three minutes later.
The corner forward beat Paddy Kenneally after being set up by Shane Kelly.
A huge free by goalkeeper Richard O’Keeffe and a brilliant point from captain Kevin Punch left them six points clear and in a comfortable position.
This was also due to an outstanding display by Shane O’Donovan, who fronted a tight defence from centre-back. Indeed, the Barrs didn’t get their first point from play until the 20 minutes from Bill Beckett.
Mayfield extended the lead to seven points, before Michael Ryan and Beckett reduced the deficit. Mayfield then found another gear. A superb line ball from Punch and a Shane Kelly point left them 1-8 to 0-4 ahead at the change of ends.
St Finbarr’s made three substitutions for the second-half, but Mayfield remained in the ascendancy. Nicky Kelly’s free-taking helping them go 1-15 to 0-8 up with five minutes remaining. Credit St Finbarr’s – who must now play Grenagh in round 2 - they did recover well with 55th and 59th minute goals from skipper Ryan.
Scorers for Mayfield: N Kelly (0-6, 0-4 frees, 0-2 65m), S Duggan (1-0), K Punch (0-3, 0-1 sl), R O’Keeffe (frees) and S Kelly (0-2 each), D O’Neill, P Duggan and K Walsh (0-1 each).
Scorers for St Finbarr’s: M Ryan (2-2), B Beckett (0-4, 0-3 frees), K Hodgins (0-2), D Quinn (0-1).
MAYFIELD: R O’Keeffe; B O’Leary, K Brosnan, P Condon; C Bond, S O’Donovan, B Punch; D O’Neill, K Punch; M J Coffey, N Kelly, S Kelly; S Duggan, K Walsh, P Duggan.
ST FINBARR’S: P Kenneally; A McCarthy, P Wiley, R McCarthy; C O’Mahony, D Quinn, P O’Brien; P Kennedy, S Callanan; C Greene, M Ryan, B Beckett; E Dennehy, K Hodgins, E Hill.
Subs: E Mulcahy for S Callinan, C Buckley for C O’Mahony, B O’Connell for E Hill (all half-time).