Gilchrist, who took four catches in the first innings of the opening Test against India, took another four in the second to overtake Healy's mark for the most dismissals by an Australian.
The pair had been tied on 395 before Gilchrist caught Wasim Jaffer off Brett Lee to take sole possession of top spot.
His four victims in the second innings of the 377-run victory on Saturday took him to 399 dismissals, just behind South Africa's Mark Boucher, who holds the world record with 406.
The 36-year-old will undoubtedly join Boucher as the only members of the 400-club when the second Test of the four-match series begins in Sydney on Jan. 2.
No other active player has more than 200 wickets.
"I'm just thrilled, really thrilled," Gilchrist told reporters. "It's not something I thought would happen.
"You don't naturally feel your name sits alongside those of (Rod) Marsh (355 dismissals) and Healy."
"To have done it in 93 Tests is a testament to the quality of fast bowling and spin bowling that I've had.
"I've had both bases covered there with the world's best," he added in reference to the retired Glenn McGrath and Shane Warne.
Gilchrist, however, may never have got the opportunity had he not made a key decision in 1994. He was considered the second-best wicketkeeper in his native New South Wales and pondered giving up the gloves altogether.
A move to Western Australia gave him an opportunity to concentrate on his glovework and allowed him to blossom into a player that has changed the wicketkeeper-batsman role.
"I had to move states in pursuit of a game for Australia, which you never think you're going to do," he said.
"You grow up in a spot and call that home and that's where you think you'll always be, so that's a pretty big decision.
"I also wondering whether to continue wicketkeeping at that time as well, I suggested to Rod Marsh perhaps I should stay in New South Wales and just play as a batsman and he just laughed me off and said 'no way'.
"Many others would say I should have taken my own advice and just given up the gloves but that's what I've loved and wanted to do.
"We all go through those little waves of rolling emotions and hopefully, if you deal with it well, you'll come out on top."
(Editing by John O'Brien)