Liverpool got the better of Premier League champions Manchester City in the International Champions Cup, winning 3-1 on penalties.
The match in New York was decided in a shootout after scores were level at 2-2 at the end of regulation time.
All four goals came in the second half and Henderson was among those on the scoresheet, with his 59th-minute equaliser cancelling out Stevan Jovetic's opener for City six minutes earlier.
It was a rare goal for Henderson, who had only managed a total of five in all competitions last season but Rodgers has tipped the 24-year-old to exceed that tally in 2014-15.
"That's the objective for Jordan [to get more goals]," Rodgers told reporters post-game.
"The finish he produced to get the equaliser, he does that day in, day out in training.
"He's a player that is so fit and strong, and he's such a team player - he does a lot of unselfish work. This year we're asking him to get more into the box.
"I thought he had more opportunities when he played just around the corner [of the striker] in the last 20 minutes of the game - he was able to time his runs better going forward and he got in a few times. That's his strength - he can really penetrate the defensive line.
"He's an athlete and can run all day. I would expect him to be getting up to 10 goals this season because he has the quality and tactical awareness to arrive into the area to do that."
Meanwhile, Rodgers said he is planning to add to his six new signings before the transfer window closes.
Liverpool have been active in the market, signing Adam Lallana, Lazar Markovic, Dejan Lovren, Emre Can, Rickie Lambert and Divock Origi, who will remain at Lille on a season-long loan.
But the 41-year-old is adamant there is still plenty of work to do upon the club's return to the UEFA Champions League.
"We still have some positions that we want to fill within the squad so that by the end of August we have a strong group of players to go forward with," he said.
"We've still got some business to do."
