Captain John Obi Mikel, a Chelsea team-mate of Moses, was also on target for Nigeria at a packed Uyo stadium while Nabil Bentaleb scored for Algeria with an unstoppable drive.
With Cameroon surprisingly held 1-1 at home by Zambia, Nigeria have taken a four-point lead after matchday 2 in Group B, regarded as the toughest of the five African mini-leagues.
Only section winners qualify for Russia and Nigeria are sitting pretty with a maximum six points from two matches, Cameroon have two and Zambia and Algeria one each.
If Nigeria win three of their remaining four qualifiers from next August -- two of which are at home -- they will be guaranteed to finish first and secure a sixth World Cup appearance.
A lucky deflection off a defender allowed Moses to score on 20 minutes and Mikel doubled the lead three minutes before half-time while Algeria wrongly appealed for offside.
Bentaleb offered hope to the visitors with a thunderous 67th-minute drive only for Moses to score again in stoppage time with a close-range shot after pouncing on a low cross.
Collins Mbesuma shook Cameroon in Limbe by finishing off a Rainford Kalaba cross after a clever free-kick manoeuvre to give Zambia a 34th-minute lead.
Vincent Aboubakar levelled five minutes into first-half stoppage time from a penalty, but for Cameroon it was two points dropped as much as one gained.
Morocco coach Herve Renard had to settle for a 0-0 draw in Marrakech against former team the Ivory Coast -- the third goalless stalemate in four Group C matches.
African champions Ivory Coast will be happier with the result as they maintain their lead with four points, Gabon and Morocco have two each and Mali one.
The Ivorians held the Moroccans with a depleted team, lacking suspended goalkeeper Sylvain Gbohouo and injured defender Eric Bailly and attacker Gervinho.
Renard, who has guided Zambia and the Ivory Coast to Africa Cup of Nations titles, may regret dropping leading scorer Youssef El Arabi.
South Africa converted a penalty that should not have been awarded as they beat Senegal 2-1 in Polokwane and trail Group D leaders Burkina Faso on goal difference.
Senegal defender Kalidou Koulibaly was judged to have handled the ball just before half-time, but big-screen replays proved that it rebounded off his legs.
Captain Thulani Hlatshwayo netted the spot-kick and Thulani Serero added a soft second goal in first-half stoppage time for a flattering two-goal advantage.
The visitors dominated the second half and halved the deficit on 76 minutes when Cheick N'Doye slammed the ball into the net from close range after two attempts were blocked.
Burkina Faso rocked Cape Verde by scoring twice within 29 minutes of the kick-off in Praia, then survived sustained second-half pressure for a 2-0 victory.
Banou Diawara struck after two minutes in the multi-island state off Senegal and Prejuce Nakoulma doubled the lead.
Burkina Faso and South Africa have four points each, Senegal three and Cape Verde are pointless.
Uganda followed up a shock draw in Ghana last month by edging Congo Brazzaville 1-0 in Kampala thanks to goal from Belgium-based teenager Farouk Miya on 18 minutes.
Victory completed a memorable year for the east Africans as they also booked a place at the biennial Cup of Nations tournament for the first time since 1978.
Uganda have four points in Group E, Egypt three, Ghana one and Congo are pointless. Egypt host Ghana in Alexandria Sunday.