The City defender confidently stroked home the Lionesses’ second goal of the night from the penalty spot after Jess Park’s strike had been handled inside the hosts’ area.
Lauren Hemp also started for Sarina Wiegman’s side in Dublin, with Chloe Kelly later coming off the bench for the European Champions.
The Lionesses had been held to a 1-1 draw in their opening Group A1 qualifier against Sweden, but cruised into an early lead at the Aviva Stadium.
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England had gone ahead through Chelsea’s Lauren James following a set piece, despite a bright start from the Republic of Ireland.
And the advantage was doubled by Greenwood soon after, seeing the reigning European champions race into a two-goal lead inside 18 minutes in Dublin.
The defender was presented with another chance from 12 yards before the half hour through another hand ball, but on this occasion her effort came crashing back off the post to keep the advantage at 2-0 heading into the break.
Park and Hemp were both replaced just before the hour as England looked to turn the screw, with Kelly introduced late on at the Aviva Stadium.
But it was the hosts who went closest in the final stages, with Hannah Hampton saving smartly from Caitlin Hayes, before the goalkeeper’s blushes were spared when Irish skipper Katie McCabe blocked her clearance, before recovering in time to make the save from a tight angle.
Nonetheless, the Lionesses held on to their clean sheet and ensured they moved up to second in Group A1, two points behind France.
Elsewhere, Kerstin Casparij’s Netherlands also grabbed their first win of Euro 2025 qualifying when they beat Norway 1-0.
The City defender played the full 90 minutes for her nation, with Juventus’ Lineth Beerensteyn handing the hosts an early lead inside six minutes, which proved the difference.
It means that Holland now sit level with Norway on three points after two matches, but sit third due to their inferior goal difference following a 1-0 defeat to Italy in their opening group match.
Finally, Yui Hasegawa and Japan had to settle for a fourth-placed finish at the SheBelieves Cup after a penalty shootout defeat to Brazil.
The City midfielder was one of the unlucky three to miss their spot-kicks as the South Americans ran out 3-0 winners after the two sides couldn’t be separated.
Mina Tanaka handed Japan the lead, but Cristiane responded with 20 to play to earn a 1-1 draw and set up for penalties.
