Local television footage showed Abbasi's motorcade arriving in the northern resort town of Muree, where Sharif is staying with his family after the Supreme Court last week disqualified him over failure to disclose a source of income.
Sharif's Pakistan Muslim League-Nawaz (PML-N) party holds a solid parliamentary majority and moved quickly to project an air of continuity after the change of prime minister.
A new federal cabinet was due to be sworn in later on Wednesday, with local media reporting that Abbasi would make the decision in consultation with Sharif and retain most of the former prime minister's loyalists.
Abbasi himself served as petroleum minister in Sharif's previous cabinet, which stepped down after he was removed.
A quick transition may ease fears that the nuclear-armed nation will be plunged into another bout of political turmoil, which could erode economic and security gains since the last poll in 2013.
Sharif's allies have dismissed Friday's ruling as a targeted campaign to unseat the popularly elected prime minister, and a member of his former cabinet implied that elements of the powerful military had a hand in his ouster.
The ruling PML-N party on Tuesday used its majority in the National Assembly to install Abbasi who is expected to serve until Sharif's brother, Shahbaz, becomes eligible to take over, probably within two months.
PML-N lawmakers banged on benches and chanted "Lion, lion Nawaz Sharif" after Abassi was voted in, standing defiant in the wake of the Supreme Court's decision to cut short Sharif's third stint in power.
(Reporting by Saad Sayeed; Editing by Michael Perry)