"The forces of the Libyan army shot down an enemy plane that was preparing to conduct air raids in the Wadi al-Rabie area", Government of National Accord spokesman Mohamed Gnounou told AFP.
Egypt has close ties with Haftar, whose Libyan National Army (LNA) controls the east and swept through the mainly desert south earlier this year before moving to Tripoli ten days ago in a major escalation of conflict.
Fighting in Tripoli has killed 121 people and wounded 561 since commander Khalifa Haftar launched an offensive earlier this month to take the Libyan capital, the World Health Organisation said Sunday.
Egypt has supported Gen Haftar's drive against Islamic militants in the east and south of its western neighbour, reportedly providing both funding and arms to his LNA.
The Libyan commander has modeled his political style of authoritarian leadership after Sissi, himself an army general turned president.
It is not clear whether Gen Haftar's surprise march on Tripoli was launched with the prior knowledge or the approval of the Egyptians.
The UN-backed GNA later issued a statement saying that it had downed a military jet belonging to the Eastern forces after it attacked the positions of its forces. They say it could also give militant groups based in Libya a cause or a motive to unite their ranks in a potentially ruinous fight against the LNA.
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Fighting in recent days has been taking place on the outskirts of Tripoli as LNA forces have been bogged down by groups aligned with the Government of National Accord (GNA).
Observers fear a civil conflict matching the 2011 rebellion that led to the ouster and death of dictator Moammar Gadhafi.
Despite the risk, the WHO said it was sending more medical supplies and health staff to Libya to cope with an anticipated increase in casualties.
"Three medical personnel have been killed and five ambulances have been incapacitated by shrapnel", OCHA said in a Saturday statement.
At least 15,700 people have been forced to flee their homes because of the conflict, according to the United Nations Office for the Coordination of Humanitarian Affairs (UNOCHA), with a "significant number" of others still stuck in live conflict zones.
"Our position will not change", United Nations envoy Ghassan Salame tweeted.