Harry Kane swept England ahead at Old Trafford before Saka stole the show with two superb finishes either side of Marcus Rashford's goal.
Saka completed his treble early in the second half, Kalvin Phillips added another and Kane then struck again from the spot as England won its fourth straight match in Group C.
"It's a feeling I can't put into words. I can't describe how I'm feeling right now. I'm so happy," Saka told Channel 4. "It's been a really long season. It's the end of the season now. I couldn't have ended it in a better way."
England tops the group on a maximum 12 points, six ahead of Ukraine, which is second after a 1-0 win over Malta in Trnava was secured by Viktor Tsygankov's second-half penalty.
European champion Italy is nine points adrift in third but has played just twice becase of its Nations League schedule.
France also kept up its 100 percent record in Group B with a 1-0 win over Greece in Paris, which propelled the World Cup runner-up six points clear of its opponent.
Mbappe saw his initial spot-kick saved by Odysseas Vlachodimos, but the France striker made no mistake with his second attempt after encroachment by a Greek defender.
Greece finished with 10 men as Kostantinos Mavropanos was sent off, having escaped red earlier when he caught Antoine Griezmann in the face with his studs and gave away the penalty.
"We did what we needed to win this match. I'm very pleased with what the players have done in this difficult period," France coach Didier Deschamps told TF1."There's less experience than there was six months ago, but there's quality."
Wales in trouble. Mikey Johnston, Evan Ferguson and Adam Idah scored as the Ireland picked up their first points of the campaign with a 3-0 home victory over minnow Gibraltar.
Wales faces a difficult task to qualify after a second loss in four days, going down 2-0 in Turkey following the dismissal of midfielder Joe Morrell late in the first half.
Hakan Calhanoglu had his penalty saved before Umut Nayir and Arda Guler struck in the final 20 minutes to keep Turkey top of Group D.
Wales is five points behind Turkey in fourth place, level with Nations League runner-up Croatia, which has two games in hand.
Armenia is second after Tigran Barseghyan converted a stoppage-time penalty to snatch a 2-1 win against lowly Latvia in Yerevan.
Finland lead a tight Group H in which the top four are separated by only two points.
Substitute Daniel Hakans bagged a hat-trick in Finland's 6-0 mauling of San Marino, the world's lowest-ranked team, with Glen Kamara, Benjamin Kallman and Teemu Pukki also on target.
Kazakhstan is level on nine points in second after winning 1-0 away to Northern Ireland, Abat Aimbetov's late goal fueling the surprise qualification push.
Slovenia and Denmark drew 1-1 in Ljubljana as Andraz Sporar opened the scoring for the hosts only for Rasmus Hojlund to level before the interval.
Valentin Mihaila stunned Switzerland with two late goals as Romania fought back for a 2-2 draw in Lucerne after Zeki Amdouni's brace had put the Group I frontrunners in control.
Israel saw off Andorra 2-1, while Belarus beat Kosovo by the same scoreline in Budapest.
© Agence France-Presse