The 31-year-old famously netted an 101st-minute leveller against Argentina at the 2022 World Cup, only for his side to lose out in a penalty shootout.
On this occasion, Weghorst marked his entrance from the substitutes' bench to seal a deserved win for Ronald Koeman's side, who were almost made to pay for wasted chances.
Adam Buksa had initially handed Poland the lead against the run of play before Cody Gakpo pulled Netherlands level before the half-hour mark.
Ahead of France squaring off against Austria on Monday, Netherlands take early control of Group D, with Poland left up against it to salvage a spot in the last 16 having failed to defeat Netherlands for the 13th successive game.
Netherlands fail to make chances count
The notable team news for Poland was Buksa getting the nod in attack with Robert Lewandowski not risked due to a minor groin issue, while Koeman opted to go with a four-player Netherlands defence, rather than the back three that he has deployed on occasions.
Oranje could have taken the lead twice during the first 10 minutes. Gakpo was first denied by a smart stop by Wojciech Szczesny, before Tijjani Reijnders sent a first-time effort wide of the target from just 10 yards out.
However, it was Buksa who put Poland in front shortly after the quarter-hour mark. An inswinging corner found the towering forward, who superbly glanced a header inside the far post.
Netherlands responded by continuing to squander chances. Virgil van Dijk was unfortunate, seeing a volley superbly kept out by Szczesny, but Memphis Depay should have at least hit the target rather than blaze over from inside the penalty area.
Their deserved equaliser arrived just before the half hour. Gakpo's effort from 20 yards would have been comfortably dealt with by Szczesny, only for the ball to deflect heavily off Jan Bednarek and find the centre of the goal.
Gakpo and Depay wasted further chances in the minutes before the half-time whistle, both sending their respective efforts off target, as Poland somehow remained on level terms when seeing Netherlands have 14 shots compared to their four.
More of the same from Netherlands...until Weghorst saves the day
The chances for Netherlands continued to go begging after the break with Depay and Gakpo combining to tee up Xavi Simons, who miscued his effort from 12 yards wide of the near post.
By the 55-minute mark, Poland head coach Michal Probierz had made three changes - a sign of his concern - and it almost paid off as Buksa was only denied a second goal through a sharp block by goalkeeper Bart Verbruggen.
Koeman made alterations of his own, one of them being Donyell Malen, and the forward's quick feet on the edge of the area teed up Denzel Dumfries, who flashed a strike narrowly wide of the far post.
While the openings were not being created with the same regularity as in the first half, Netherlands should have gone ahead when Stefan de Vrij was presented with a free header inside the area, only to send it sailing over the crossbar.
A different approach was taken with 10 minutes remaining, Koeman hoping that the extra height of Weghorst would influence proceedings, but it was the composure of the frontman that got him on the scoresheet.
Nathan Ake bided his time before threading a pass inside to the former Manchester United man, who swept the ball home inside the near post from 10 yards out.
Poland pushed admirably for an equaliser, with Verbruggen pulling off a brilliant save to deny Karol Swiderski, but the late assault did not pay off as Netherlands secured the three points that their performance warranted.
When do Poland, Netherlands play again at Euro 2024?
All eyes will now be on France's encounter with Austria on Monday before the second batch of games take place on Friday.
The 5pm kickoff sees Poland go head to head with Austria, before Netherlands and France meet in a blockbuster clash on the evening.