The Hammers picked up only their second victory of the new campaign last Saturday, beating the Merseysiders 3-1 at Upton Park.
With West Ham now in the top half of the Premier League table above the likes of Liverpool, Everton and Manchester United, Jenkinson believes that the team can have a "great season" if they continue to perform to the same level that they did on the weekend.
"Everyone is pulling together for the same cause and wants to do well for the club and we want to have a strong season," he told West Ham's official website.
"If we continue how we started on Saturday and we can carry that on, we can be in for a great season. I am just focusing on the next game now and hopefully we can all take that into the weekend."
West Ham continue their Premier League campaign on Saturday when they travel to Manchester United.