The Nicaraguans recorded a shock 3-2 win in Kingston last week and were going through on away goals until Dawkins struck with almost the last kick of the game.
It was the biggest result of a busy evening in the CONCACAF region of North, Central American and the Caribbean.
Twelve teams were competing for six spots in the penultimate stage on the road to Russia 2018.
Among the other victorious teams were Haiti, who beat Grenada 3-0 to go through 6-1 on aggregate, and Canada, who qualified 4-1 on aggregate after drawing 1-1 in Belize.
A goal six minutes from time was enough to give Saint Vincent and the Grenadines a one-goal aggregate win over Aruba and they will be joined in the next round by El Salvador who beat Curacao 1-0 to record a 2-0 aggregate victory.
Guatemala join them after their 2-0 win against Antigua and Barbuda allowed them to overturn a 1-0 deficit from the first leg.
The six winners now go into next month's group stage alongside seeds Mexico, Honduras, Panama, Costa Rica, the USA and Trinidad and Tobago.
The teams will be divided into three groups of four and each team will play the others in their group home and away with the top two in each group progressing to the fifth round.
The top three teams in that group of six qualify automatically for Russia in 2018, while the fourth-placed side will play off against a team from Asia for a final berth.
(Writing by Andrew Downie; Editing by Amlan Chakraborty)
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