The former England boss took charge at Selhurst Park during the festive period and has eased the club's relegation fears with three consecutive wins prior to the international break.
Palace have not conceded a Premier League goal since February's defeat to Stoke and could secure a top-10 finish if they can maintain their form during the remainder of the campaign.
Hennessey has been in goal for Palace's trio of shut outs and is delighted to have secured regular game time since Allardyce's arrival in south London.
"When a new manager comes in every player wants to try and keep their shirt," he told the Sun. "Hopefully that's what I've done with Sam Allardyce coming in.
"Sam came in and settled really well - he's got us firing. It's always nice to have competition in Julian Speroni and Steve Mandanda.
"I just need to keep working hard. At club level I'm on the back of of three wins, three clean sheets so hopefully that can continue," he added.
A trip to Stamford Bridge to face the Premier League leaders is expected to be a stern test of Palace's recent improvement.
Antonio Conte's side are in superb form and will be determined to maintain their enviable position at the top of the table with victory over their capital rivals.
And Hennessey, who has been on the receiving end of some spectacular goals this season - including Olivier Giroud's scorpion flick - says he can do little about such outstanding finishing.
"You don't mind fantastic goals going in against you but there was nothing I could do.
"Some of the goals that were going in against me were unstoppable," added the Welshman.