The visiting captain's spot-kick in the 90th minute cancelled out Owain Tudur Jones' 30-yard thunderbolt opener, which came in the 19th minute at Tulloch Caledonian Stadium.
The draw extended Inverness's unbeaten run to 10 games in all competitions.
It is likely to do little to ease the woes of Hearts, who might have played their last away game if the club succumbs to a winding-up order over a tax bill of almost £450,000.
The visitors made two changes to the side which lost 1-0 at Dundee with Ryan Stevenson and Jason Holt in for John Sutton, who was dropped to the bench, while Scott Robinson was not included in the squad.
Richie Foran was restored to Caley Thistle's starting line-up in place of Ross Draper, who failed to overcome an ankle injury he suffered in stoppage time of last Saturday's 2-1 win at Kilmarnock.
Inverness started brightly with top scorer Andrew Shinnie connecting acrobatically with an Aaron Doran cross to volley the ball on target only to be denied by diving Hearts keeper Jamie MacDonald.
The visitors responded with Stevenson forcing Antonio Reguero into his first save on the quarter-hour mark. But four minutes later Inverness took the lead through a 30-yard rocket from Tudur Jones.
The Welshman hammered the ball past MacDonald for his first goal of the season after a lay off from Andrew Shinnie.
Inverness might have added a second on the half-hour when Billy McKay ghosted through the Hearts defence, but the striker's low shot went just wide.
MacDonald pulled off another diving save to thwart Inverness seconds before half-time, and the shot-stopper kept out a 20-yard effort from Foran after nifty footwork from McKay to set him up.
Andrew Shinnie might have put Caley Thistle further ahead 10 minutes after the re-start after being played in by Doran, but MacDonald stopped was equal to the attacker in a one-on-one situation.
The ball rebounded to Tudur Jones but his shot was blocked by Zaliukas.
Stevenson remained Hearts' most dangerous player and had two shots from the edge of the area blocked by Inverness centre-back Gary Warren.
Full-back David Raven forced another save from MacDonald when his shot almost found the top corner.
Substitute Sutton headed over from a Danny Grainger cross as pressure from the visitors continued to build.
It paid off in the 90th minute when Zaliukas scored the leveller from the spot. Referee Willie Collum awarded the penalty after Callum Paterson appeared to be fouled in a crowded area.
- Sports & Recreation
- Owain Tudur Jones
- Andrew Shinnie
- Ryan Stevenson