Having scored 11 goals without conceding in their opening two Pool A victories everything seemed to be going to plan again at the interval.
Matt Butturini picked up a penalty corner rebound to open the scoring in the 11th minute and then Dwyer added his record-breaking goal in unspectacular style a few seconds before half-time.
But Argentina, who have only ever beaten Australia once before at the Olympic Games in 1976, produced a rally after the break on the back of some impressive goalkeeping by Juan Manuel Vivaldi.
Captain Matias Vila drilled in a shot from the edge of the circle which took a deflection off defender Fergus Kavanagh to reduce the deficit and then Lucas Cammareri had a close-range shot blocked by Nathan Burgers.
But Australia stepped up a gear and Kieran Govers hit a post, Dwyer fired over after a brilliant first-time cushioned pass by Matt Gohdes and Glenn Turner rolled a shot wide past advancing goalkeeper from Butterini's through-ball and also had another effort saved by Vivaldi.
With two minutes remaining, Argentina won a penalty corner and Gonzalo Peillat converted to earn the South Americans' first point of the tournament.
'To be the highest goalscorer for Australia is a great honour for me personally,' Dwyer said.
'I've been asked a lot about it over the last few days but it hasn't worried me, it hasn't been in the back of my mind.
'I just want to do my job for my country.'
The 33-year-old admitted, however, they had slipped up in not seeing out victory.
'It was disappointing not to win having been 2-0 up at half-time,' he added.
'But we gave them chances and they took them and we walk away with 2-2.'