Accies took the lead in the 24th minute at McDiarmid Park on Saturday when on-loan Dundee striker Mickael Antoine-Curier bundled in at the back post following Richard Hastings' deeporner.
However, Morais ensured the visitors' advantage lasted only five minutes when he headed in Kevin Moon's cross from the right.
Saints pressed hard for the winner in the closing stages but their Lanarkshire opponents held out for a precious point in a frantic finale to move them up a place to 10th spot.
Accies came into the game knowing that anything less than victory could see them drop to the foot of the SPL, but it was home side St Johnstone who began the brighter.
The Perth men forced a corner after only 16 seconds and, although that came to nothing, Kenny Deuchar worked the game's first effort on target only two minutes later as Tomas Cerny held the marksman's header from close range.
Hamilton forced two corners in quick succession in the seventh minute but could not capitalise on Richard Hastings' tempting near-post deliveries and seven minutes later James McArthur's deflected effortfell kindly for Saints keeper Graeme Smith.
Another Hastings flag-kick from Accies' right caused the home defence all sorts of problems but Simon Mensing could not direct his near-post header from close range on target.
And there was no surprise when it was from another Hastings corner that Accies took the lead in the 24th minute.
The Canadian curled a sweet set-piece into the heart of the six-yard box and when it was flicked on through a crowded goalmouth Antoine-Curier pounced to volley in from close range.
Accies' lead was to last only five minutes, however, as Saints dragged themselves back level in the 29th minute.
A neat move down the left was broken up just as Deuchar looked to pull the trigger but the McDiarmid Park outfit kept up the momentum and the ball was ferried to the right side of the box where Moon lurked.
The midfielder brilliantly picked out Morais arriving in the box 10 yards out and the winger's header flew high past Cerny for the equaliser.
The goals were brief bright points in a largely dull encounter and it took until five minutes before the interval until Sats threatened again.
Antoine-Curier made a hash of clearing a corner but when Collin Samuel redirected the ricochet goalwards the Frenchman was on hand to make amends with a goal-line clearance.
Seconds later Morais deflected a shot from team-mate Kevin Rutkiewicz and Cerny was relieved to see the effort drift over his bar.
Murray Davidson brought out a magnificent save from Cerny with a shot from 20 yards which was arrowing towards the Accies keeper's bottom right-hand corner.
Saints forced another corner from the resulting flag-kick and this time, after Rutkiewicz had just missed Morais' delivery, Deuchar lofted his back post effort on to the roof of the net.
With the home side continuing to dominate Davidson sent a header wide in the 53rd minute and seconds later Morais' shot also drifted past.
Accies manager Billy Reid had to be spoken to by referee Willie Collum after being involved in a touchline altercation with McDiarmid Park number two Tony Docherty and the bust-up came just as his side created their first opening of the second period.
Hastings' cross from the left in the 61st minute fell kindly for Antoine-Curier but Rutkiewicz got across to deflect the Frenchman's shot over.
Five minutes later home striker Samuel broke free of his markers but, despite his eventual shot moving in the air, the drive was held on the line by Cerny.
As befitting a game between two sides scrapping for points at the wrong end of the table, the game too often became bogged down in a packed midfield and referee Collum's whistle too frequently broke up play.
St Johnstone looked to have the greater impetus as they sought the winner but Morais' drive which was held by Cerny was the closest the home side came to sealing the win in the closing minutes.