Saints rejected a bid from the Hammers, which fell well short of their valuation for Harwood-Bellis, on Friday evening, before naming him in their side at Stoke.
He was the only one of Southampton's three players drawing transfer interest in the squad, with Adam Armstrong and Cam Archer left out of the side.
Harwood-Bellis, a one-time England international, showed his quality early on create the opening goal with an incisive pass to Tom Fellows, who teed Finn Azaz.
"Taylor is an ultimate professional. I think you don't find many like him in the game. The quality to block the noise at the moment, I think he was outstanding today," said Eckert.
Saints have rejected a bid from West Ham for Taylor Harwood-Bellis. (Image: Matt Watson / Southampton FC)
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"The first pass he played for Tom, there aren't many in the league that can do that. He's an outstanding player."
Eckert is not the decision maker for Saints in the transfer window, with Johannes Spors and the board making the ultimate decision whether to sell a player.
Asked how much he knows about interest, Eckert added: "I do hear a little bit every here and there, but at the same time, it's just so important that we win football games.
"Luckily, there are some people in this club higher up who are getting involved and just let me do my job."