Two of the pre-tournament favourites faced off at Bramall Lane, and it was Sweden, unbeaten so far in 2022, that took a 1-0 lead into the break after a low Kosovare Asllani cross found Jonna Andersson to slot a 36th-minute opener.
Netherlands defeated the Swedes in the semi-finals of the 2019 FIFA World Cup, and it equalised in the 52nd minute when a fortunate deflection ran into the path of Roord, who fired into the bottom-left corner.
Sari van Veenendaal in the Netherlands goal was forced off because of injury in the first half, and substitute goalkeeper Daphne van Domselaar made an important stop to deny Fridolina Rolfo a late winner.
Portugal came from two goals down to finish 2-2 with Switzerland in Leigh, in the day's other Group C game.
Switzerland had not won in six matches prior to its opener, including 4-0 and 7-0 thrashings from England and Germany respectively in pre-tournament friendlies.
But its UEFA Euros campaign looked to be off to a flyer when a 25-yard shot from Coumba Sow and a powerful header from Rahel Kiwic put the Swiss 2-0 up after just five minutes.
Portugal had not had a shot on target by the time the referee blew the half-time whistle, and needed to improve if it was to find a way back.
It did exactly that and halved the deficit after 58 minutes when Diana Gomes met a corner from the left. Her initial header was saved, but she made no mistake with the rebound to make it 2-1.
The Portuguese then got themselves on level terms just seven minutes later, when a superb cross from Tatiana Pinto picked out Jessica Silva in the box, and she smartly side-footed into the bottom corner.
Both teams hit the woodwork late on in their pursuit for a winner, with Geraldine Reuteler and Telma Encarnacao the players going close.
In the next round of fixtures on Thursday (AEST), Netherlands plays Portugal in Leigh as it looks to get a foothold in Group C, while Switzerland and Sweden will face off at Bramall Lane.