Wood scored twice in the first half while Kosta Babarouses gave the home side a 3-0 lead at the break before Solomons' captain Henry Fa'arodo converted a second half penalty to give the visitors a potentially valuable away goal.
Ryan Thomas, however, re-established the three-goal cushion before Michael McGlinchey slotted a free kick past Phillip Mango. Wood grabbed his hat-trick in added time to give New Zealand a comfortable advantage heading into the second leg in Honiara on Sept. 5.
The winner of the two-legged tie will meet the fifth-placed South American side, currently two-times World Cup champions Argentina, in an inter-continental playoff in November.
"I'm just pleased with the result and the way we applied ourselves," said Wood. "We came out and produced a professional performance and it showed in the scoreline which was nice.
"It's all about getting the result. We need to do a professional job, work from the back and take our chances when they come. We could have had a few more but we are very pleased with what we have done."
Wood, who scored his first goal for English Premier League side Burnley last Sunday before jumping on a plane to New Zealand, could have finished with four goals but for two brilliant saves from Phillip Mango.
The Solomons, who had said they would defend deep to try to stop New Zealand from scoring heavily before they head to the tropical heat and humidity in Honiara, rarely pushed past the midfield.
Their defensive tactic, however, failed as the All Whites stretched the play wide at every opportunity and then exploited the gaps with runs from midfield, while the visitors made numerous errors in defence.
Wood's first two goals came from defensive errors allowing him one-on-one opportunities with Mango in the 18th and 36th minutes, while Babrarouses powered home New Zealand's third in the 39th when Nelson Sale slipped over.
The visitors capitalised on one of their rare forays into New Zealand territory when Benjamin Totori earned a 52nd minute penalty that Fa'arodo calmly converted past Stefan Marinovic.
Thomas, however, exploited some more lax defending less than four minutes later before McGlinchey and Wood's hat-trick goal virtually assured New Zealand of advancing to the inter-continental playoff.
(Reporting by Greg Stutchbury in Wellington; Editing by Amlan Chakraborty)
