Billy McKay had put Caley in front with a close-range angled shot in the 74th minute, moments after Andrew Shinnie missed a penalty, but Robertson levelled in the second minute of stoppage time with a point-blank header.
The draw ended the Perth side's six-game winning streak in all competitions but moved them to within two points of SPL leaders Celtic.
Caley Thistle remain fifth and are five points off top spot.
Both sides were unchanged from their last outings, where Inverness won 4-1 at Dundee and St Johnstone beat Kilmarnock 2-1 at home.
Spanish goalkeeper Antonio Reguero was preferred to Ryan Esson, who returned to the bench after being sidelined for the last three games.
He was called into action early by former Inverness striker Gregory Tade but the French forward squandered the one-on-one chance.
St Johnstone keeper Alan Mannus was also called into action early, saving low from Ross Draper.
Chances were at a premium but Draper forced another save from Mannus when he headed a Philip Roberts cross on target.
Liam Craig fired wide for the Saints before Rowan Vine's shot was saved by Reguero to ensure it was goalless at half-time.
The visitors were first to carve out a chance after the interval when Nigel Hasselbaink played through Gary Miller but he was thwarted by Reguero.
At the opposite end full-back David Raven produced a dangerous cross which was turned wide by Steven Anderson before Draper or McKay could connect.
Inverness might have made the breakthrough with 20 minutes to go when an Aaron Doran pass found Graeme Shinnie who was tripped in the box by Vine, but Andrew Shinnie blazed the spot-kick high over the crossbar.
But four minutes later the hosts took the lead through McKay.
Andrew Shinnie found Draper, who flicked the ball on to McKay and the forward buried an angled shot from eight yards, his fifth goal of the season.
McKay would have doubled the lead moments later with a curling effort but Mannus performed a terrific diving one-handed save.
Andrew Shinnie fired over from four yards after Graeme Shinnie's corner was headed into his path by Owain Tudur Jones.
Saints substitute Robertson found Craig late on but he sent the chance inches over and Andrew Shinnie put another shot off target with time running out.
But the miss proved costly for the hosts as Robertson headed past Reguero from close range in the 92nd minute after a punt upfield from Mannus, ensuring it was honours even.