The African champions went to the locker room with a three-goal lead thanks to efforts from Mohamed Zidan, Hosni Abd-Rabou and Mohamed Abou-Treika.
Egypt captain and Anderlecht playmaker Ahmed Hassan completed the rout after the break with a trademark penalty-kick.
Soliman scored the consolation goal for the visitors from the spot in the 78th minute.
Egypt now top Group B with three points before Botswana play Mauritania in the group's second tie.
Egypt's players and coaching staff wore black armbands in grief over the death of Ahli's international left-back Mohamed Abdul-Wahab who died in training on Thursday.
The hosts staged a bold attacking performance against Burundi in the first-half and managed to break the deadlock as early as the fifth minute after a quick combination between Hassan and Zidan.
Bremen's Zidan received Hassan's quickly-taken free-kick before sidestepping a defender inside the area and firing home.
This was Zidan's first goal with the Egyptian national team.
Ismaili midfielder Abd-Rabou doubled the lead in the 29th minute when his deflected free-kick eluded the goalkeeper and settled into the back of the net.
In the 39th minute, Abou-Treika controlled an inch-perfect cross from Ahmed Fathi before darting the ball past the advancing keeper.
Hassan, who put on the number 3 shirt of Abdul-Wahab, widened the gap seven minutes into the second period when he pinned a low penalty into the left corner of the net.
Strasbourg left-back Ahmed Abou-Mosallam stepped over Soliman in the area and the player executed the resulted penalty past goalkeeper Essam Al-Hadari to give his side some respectability.
نرشح لكم
اتحاد الكرة يعلن إقامة مواجهة مصر وإسبانيا يوم 31 مارس
الأهلي يعلن استدعاء ديانج وكامويش لمنتخبي مالي وكاب فيردي
بتواجد كافة أعضائه.. انطلاق الاجتماع الطارئ لاتحاد الكرة
كامويش في قائمة كاب فيردي لأول مرة استعدادا لتوقف مارس
وزارة الشباب والرياضة تكشف تحركاتها لعدم تأثير إجراءات التقشف على أداء المنشآت
ممدوح عيد: سلامة لاعبي بيراميدز أهم من أي بطولة
جراحة ناجحة في الرباط الصليبي لـ إبراهيم الأسيوطي لاعب شباب الأهلي
كأس مصر – زد إلى النهائي للمرة الثانية في تاريخه بعد ثلاثية ضد طلائع الجيش











