Anand notched up an emphatic win over American Marc Esserman, while Harika beat Englishman Oyama Akito in the first round here late last night.
Playing with white pieces, Anand showed much better precision than his opponent to eventually complete the proceedings in 46 moves.
Anand will next take on Germany's Lampert Jonas in the second round.
The World No. 10 Harika, also playing with white pieces, started off the game on an aggressive note against her male opponent and made breakthroughs from the initial phases of the game.
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"I found myself in a good position from the start of the game and didn't let the pressure down on my opponent which eventually gave me an easy win," Harika conceded after her first-round game.
She will now take on the German FIDE Master Babar Michael in her second-round game and will look to continue her momentum buoyed from the first round.
"I have no special preparations for this game and I am looking to take it one at a time," she quipped before her game.
The Isle of Man International Chess Tournament features the crme de la crme of the chess fraternity, with the World numero uno, Magnus Carlsen participating alongside the likes of Caruana Fabiano, Nakamura Hikaru, Short Nigel D among others for the handsome cash prize on offer.
The Open tournament is being played at a time control of 100 minutes for 40 moves, followed by 50 minutes for 20 moves, then 15 minutes for the remaining moves with 30 seconds added per move from the start.