After three straight losses to start the league campaign, Slaven Bilic's side claimed all three points from their home encounter with Huddersfield Town courtesy of second-half strikes from Pedro Obiang and Andre Ayew.
For Hart, on loan with the Hammers from Manchester City, the game also marked his first clean sheet in 23 league games, a run stretching back to his time with Torino in Serie A last term.
"It's about time," the England stopper told The Telegraph. "I have kept plenty for my country. Whether we keep a clean one or not, as long as we win I am not too interested.
"We stuck to the plan and fully deserved the win. We are up and running and it's nice to be at home and get a result. We got a got a slice of luck for the goal but it was coming.
"We had not deserved to win any games so we were where we deserved to be.
"You can't let people outside dictate how you feel but we knew that it was not good enough what we were doing and that we needed to perform."
Next up for West Ham is a trip to West Bromwich Albion on Saturday afternoon.