Newcastle had lost their previous three games, reviving jitters they may be dragged back into the battle against the drop as the season heads towards its finale.
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But with the hosts dominating a surprisingly toothless Wolves, New Zealander Woods cleverly drew a foul from goalkeeper Jose Sa as he chased a Joelinton pass, before confidently thumping home the spot kick in the 72nd minute.
"I'm so pleased for him. He's a really willing lad. That will do wonders for him," manager Eddie Howe said of his January signing.
"We still have work to do and it brings us closer to safety, but we're not there yet," added Howe, warning against complacency even though Newcastle now look reassuringly-positioned in 14th on 34 points with seven games remaining.
Wood, whose former club Burnley are third from bottom, was thrilled to have broken his home duck and further banish the relegation spectre.
"It is fantastic. There is something about playing in front of St James' Park and we want to get them up for it. Luckily today we did," he said.
"It's nice to get my first goal here but ultimately we need points and to get out of the relegation battle, so I'm more delighted with the three points. We want this club to go to the next level. We need to get to 40 points and then build."
Wolves missed the suspended Raul Jimenez and did not even manage to touch the ball in Newcastle's penalty area in the first half. Only once, 10 minutes before the end, was Newcastle goalkeeper Martin Dubravka forced into action, diving low to save a Fabio Silva shot.
Defeat left Wolves eighth on 49 points, their outside hopes of a berth in next year's Champions League receding.
"We weren't aggressive, we didn't create our chances," disappointed manager Bruno Lage said of his team's insipid first half.
The margin of victory could have been bigger for Newcastle: Woods had an earlier goal disallowed, winger Allan Saint-Maximin shot over, and midfielder Bruno Guimaraes also went close in an overall man-of-the-match display.