Callum Hendry had given Saints an early lead but – after coming from two down to beat Braga in midweek – Steven Gerrard's side launched another comeback.
May's goal made it 10 points Rangers have dropped in eight matches since returning from the winter break.
The away side were looking to use their European exploits to kick-start their faltering title charge but they made a sloppy start.
Niko Katic hit a half-chance wide and Hendry had a decent strike saved before the pair stayed on centre stage for the eighth-minute opener.
The Rangers centre-back could not deal with an up-and-under down the left and Hendry brushed him off before taking a good touch that saw him bearing down on goal. The forward – the son of former Gers defender Colin Hendry – showed impressive composure to guide the ball through Allan McGregor's legs.
Rangers continued to look ragged in defence and McGregor saved from Matt Butcher after a mistake from Connor Goldson.
Zander Clark stopped decent efforts from Ryan Kent and Ryan Jack before Saints missed a good chance when Jamie McCart could not keep a free shot down from 16 yards.
The hosts were defending well and snuffed out good chances for Scott Arfield and Alfredo Morelos. Ianis Hagi, who masterminded the midweek comeback, was looking lost on the heavy McDiarmid Park pitch and he made a mess of a shot inside the box.
Rangers' best first-half chance came when Clark punched Kent's cross straight to James Tavernier but the Saints goalkeeper made amends with a point-blank stop.
Gerrard's half-time switch saw Kamberi replace left-back Andy Halliday. Aribo went to left-back – as he did against Braga – in a new 4-4-2 formation.
Arfield shot over early in the half but Saints squandered a great chance to double their lead when Drey Wright dribbled past Katic but saw his shot blocked by Goldson.
The equaliser soon came in the 50th minute. Tavernier's diagonal ball bounced invitingly for Kamberi on the edge of the box and the on-loan Hibernian forward showed brilliant technique to volley into the top corner.
The chances continued at both ends. Kent set up both Tavernier and Kamberi but the former fired well over and the goalscorer drew a good stop from Clark.
McGregor was also called into action to divert Anthony Ralston's dangerous ball across the face of goal.
Kamberi engineered Rangers' second goal in the 71st minute with a dummy to turn Jason Kerr. He then sent in a low cross which was stabbed home by Aribo.
Morelos was wide with a good chance before being replaced by Steven Davis as Gerrard responded to Tommy Wright throwing Michael O'Halloran on up front.
St Johnstone soon levelled when May drilled home from eight yards after David Wotherspoon's corner eventually landed at his feet.
Rangers threw men forward including Katic but they could not fashion a clear chance for a winner.