The visitors, who have already secured the league title, looked the more likely in the opening 10 minutes as Kieran Tierney was on the attack on his full debut for Celtic, while James Forrest saw his shot blocked.
Michael O'Halloran was involved in a number of chances for the hosts, but he saw a shot turned behind for a corner, before Ambrose cleared his header from the resulting set piece off the line.
Brown, Kris Commons and Griffiths all linked up twice in the space of a couple of minutes as Celtic tried to edge ahead down the other end, but Commons was twice unable to connect to get the shot away.
The Hoops had a free kick with just over 20 minutes remaining as they continued to look for a way to break the deadlock, but Mannus denied Charlie Mulgrew as the defender tried to head in from Stuart Armstrong's delivery.
St Johnstone still threatened on the counter as the final 10 minutes approached though, with O'Halloran forcing Zaluska to tip his effort behind for a corner, which the home side could not make the most of.
Brian Easton tried his luck on the volley from the edge of the box late on in a bid to snatch victory for the Perth side, but Ambrose was on hand to block the strike for the visitors.
Liam Caddis also tested Zaluska in the closing stages of the match, rifling the ball in low towards the Celtic goal, but the keeper was ready again, ensuring that the sides finished level.
The result increases Celtic's lead at the top of the table to 15 points, while St Johnstone remain third in the table.