The Reds went into the game having failed to beat Pulis in the league since January 2011 - a run stretching eight games and spanning the Welshman's spells as manager of Stoke City, Crystal Palace and now the Baggies.
But that stubborn unbeaten streak was ended as Sadio Mane and Philippe Coutinho both scored before the break to establish a comfortable lead.
That advantage should have been added to in the first half-an-hour of the second-half as Liverpool continued to dominate, but they failed to do so and Gareth McAuley poked home with nine minutes left to set up a nervy finish.
Liverpool had yet more chances to put the game beyond doubt in the dying stages, but Jürgen Klopp's men clung on for all three points and Clyne was delighted by the performance of his teammates, particularly those in the final third.
Defender insists West Brom 'couldn't live' with them
"It was a very professional performance," the right-back told Liverpoolfc.com as he insisted that the hosts "knew what we had to do" in order to "get the three points."
Clyne felt that Klopp's side "played really well" and highlighted their first-half performance, in which he felt Liverpool "took the game" to West Brom and were "really bright" which meant the visitors "couldn't live with us."
The England international acknowledged that Liverpool were "disappointed not to kick on in the second-half" but praised the team for the way they "stuck in there and managed to defend what they threw at us."
The 25-year-old also said that the Reds were "happy with the three points" as he suggested that they came away slightly frustrated having not more goals past West Brom.
He called it "important" to have broken the deadlock after 20 minutes, insisting that the "early goal" meant Pulis' side "pushed out a bit more" which gave Liverpool "more space."
Reds could have scored 'four or five', says Clyne
Clyne said it was "crucial" to get the goal early on and also continued: "It's good that we're creating a lot of chances, we've got a lot of creativity up front."
He added that they "created a lot of chances" again and that "on another day, it could have been four or five."
The defender noted Liverpool's fluidity, insisting they had "a lot of movement up front" and "a lot of ability on the ball and off the ball and a lot of chances", saying that it is "good to play in a team that's creating chances."
On what their win means for their start to the season, Clyne said: "We're very pleased. It's a great start. We're second at the moment and we're pushing on."
He explained that they are "doing well" but that they are only "taking each game as it comes" and "trying to win the next game and the game after" and then "keep going" after that.