Frazer Wright made a goal-line clearance in the opening minute to prevent Saints' Collin Samuel scoring.
Further unsuccessful efforts came in from Collin Samuel and Grainger as Saints continued to probe.
May, 18, put the Perth Saints ahead after only four minutes, scoring from Collin Samuel's cut-back.
Saints had the better openings after the break and Collin Samuel missed an excellent chance to put his side ahead when he shot tamely at Mark Brown.
The close season saw McInnes add Kenny Deuchar, Murray Davidson, Danny Grainger and Filipe Morais to a squad that contained a few experienced campaigners like Jody Morris, Paul Sheerin and Collin Samuel.
After 20 minutes, nice play on the wing by Collin Samuel saw the ball eventually fall to Jamie Adams, but the Saints midfielder hastily fresh-aired his shot and a decent chance was gone.
Jennison Myrie-Williams earned a start for St Johnstone in place of the injured Murray Davidson, while Marcus Haber, Peter MacDonald and Steven Milne came in for Collin Samuel, Sam Parkin and Andy Jackson.
Moments later, a low drive from Jim Goodwin was easily gathered by Saints keeper Graeme Smith, then Collin Samuel fired over the Accies bar from just outside the six-yard box at the other end.
Collin Samuel came on but his side almost lost their lead immediately when Lynch lobbed the ball into the six-yard box following a half-cleared corner, however Enckelman stopped Lee Mair's header right on his line.
McInnes freshened up his side after the midweek Co-operative Insurance Cup defeat, Jennison Myrie-Williams coming in for the injured Murray Davidson, while Marcus Haber, Peter MacDonald and Steven Milne came in for Collin Samuel, Sam Parkin and Andy Jackson.
BBC: Saints boss McInnes unhappy with basic defensive errors
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