The attacker made only his fifth Premier League start of the season as the Reds swept aside the Cherries 4-0 at the Goldsands stadium.
Goals from Philippe Coutinho, Dejan Lovren, Mohamed Salah and Roberto Firmino secured the win which keeps Liverpool fourth in the table, one point above Arsenal.
But it was the performance of Oxlade-Chamberlain which caught the eye of Jenas.
The former Tottenham and Newcastle midfielder claimed that Oxlade-Chamberlain needed the first half of the season to settle in at Anfield, but he has backed him to now flourish in the coming months.
“He is probably the most comfortable that I’ve seen him in the shirt,” Jenas told Match of the Day.
“He looks like he’s starting to get to know his team-mates a lot better which is a massive part of starting to enjoy your game and playing the way that you want to.
“He had an intensity about his game today.
“He got about Bournemouth and didn’t allow them to have a second.
“The confidence he had was great. He was a thorn in the side of Bournemouth.
“He was always on the front foot and trying to make things happen.”
Oxlade-Chamberlain will now hoping to retain his starting spot against Arsenal on Friday (7.45pm).