Arjun dismissed Bhuvneshwar Kumar for two runs in the final over of Tuesday's match, which Mumbai won by 14 runs. He conceded 18 runs in 2.5 overs.
The Tendulkars became the first father-son duo to have played in the IPL after Arjun made his debut on Sunday against former champions Kolkata Knight Riders.
Rohit said Arjun, who was picked by Mumbai during the auction in 2021, was confident about what he wants to do for the team.
"We saw in this leadup to the IPL what he was doing - he was bowling fast, nailing those yorkers and quite clear in his plans as well," Rohit told reporters.
"Keeping things easy - that's what he is trying to do."
Arjun, whose father Sachin is part of Mumbai's coaching staff, said they discuss tactics before matches.
"He tells me to back what I practise every game," Arjun said. "That's it."
