Other African powerhouses had mixed fortunes with South Africa snatching a last-gasp 3-2 win in South Sudan while Nigeria were held 0-0 in Rwanda.
Jean-Philippe Krasso, a late addition to the Ivory Coast squad, opened the scoring as the defending champions beat Chad 2-0 in Yaounde to lead Group G by three points from Zambia.
East Africa had cause for celebrations with Kenya beating Namibia 2-1 and Tanzania edging Guinea 2-1 to collect maximum points from away outings.
Travel difficulties meant Botswana arrived in Francistown from Mauritania only seven hours before facing record seven-time AFCON title-holders Egypt, captained by Salah.
The home team fell behind after only four minutes when a Salah shot was parried by goalkeeper Goitseone Phoko and Mahmoud Hassan - popularly known as Trezeguet - netted.
Trezeguet struck again to give the visitors a 2-0 half-time advantage that was increased on 56 minutes when Salah tapped in a cross that evaded four defenders.
Mostafa Fathy completed the rout in added time to give Group C leaders Egypt a maximum six points, having launched their campaign last Friday with a 3-0 win over Cape Verde.
Goalkeeper Fiacre Ntwari starred for Rwanda as they held 2024 AFCON runners-up Nigeria before a vibrant Kigali crowd that included President Paul Kagame.
Ntwari repeatedly defied the Super Eagles, especially Ademola Lookman, the lone African among the 30 nominees for the 2024 Ballon d'Or men's award.
Lookman claimed a hat-trick when Atalanta defeated Bayer Leverkusen 3-0 in the UEFA Europa League final in Dublin this year.
When the Nigerian star did get the ball in the net in Kigali, from a header midway through the first half, the goal was disallowed because a teammate had committed a foul.
Rwanda are 92 places below Nigeria in the world rankings, but their commitment and energy were commendable, and the Wasps deserved the point.
Nigeria have four points, Benin three, Rwanda two and Libya one in Group D. The next assignments for the Nigerians are home and away fixtures against the Libyans in October.
Substitute Thalente Mbatha was the toast of South Africa again as he scored the 95th-minute winner against minnows South Sudan on an artificial pitch in Juba.
When the home side only partially cleared a Teboho Mokoena free-kick, Mbatha rifled the ball past a throng of players and Australia-based goalkeeper Majak Mawith into the corner of the net.
After falling behind to a penalty converted by Toto Okello, South Africa led 2-1 at half-time thanks to an Oswin Appollis brace.
Sloppy defending enabled Valentino Yuel to equalise on 57 minutes, and the Bright Stars looked set to collect an unexpected Group L point until Mbatha struck.
Mbatha made his national team debut last Friday off the bench and also scored five minutes into added time, enabling South Africa to snatch a 2-2 home draw with Uganda.
Uganda top Group K with four points, ahead of South Africa on goal difference. Congo have three points and South Sudan none.
A brilliantly struck free-kick by John Avire set up Kenya for victory over Namibia in Soweto, and first place in Group J, above five-time African champions Cameroon on goals scored.
Mudathiri Yahya struck on 88 minutes to grab a 2-1 win for Tanzania over Guinea in Yamoussoukro. The Taifa Stars lie second in Group H, two points behind the Democratic Republic of Congo.
