Argentina had much the better of the first half and mercurial captain Messi put it in front with a brilliant free-kick in the Group B encounter at Estadio Nilton Santos.
Vargas came to Chile's rescue after the break, heading home the rebound after Emiliano Martinez saved Arturo Vidal's penalty.
Nicolas Gonzalez was particularly wasteful as Argentina was unable to secure what would have been a deserved victory in Rio de Janeiro.
Messi is yet to win a trophy with Argentina after back-to-back runners-up performances at the Copa America in 2015 and 2016, having also fallen short in the 2007 final against Brazil.
Giovani Lo Celso was Chile's chief tormentor in a promising start from the Albiceleste, setting up chances that Lautaro Martinez and Gonzalez were unable to take.
Gonzalez headed over the crossbar after his shot was palmed away by Claudio Bravo and the forward also failed to beat the Chile goalkeeper with a tame finish when the lively Lo Celso sent him clear with an incisive pass.
Messi produced yet another moment of magic to put Argentina in front, though, bending a brilliant free-kick into the top-right corner after 33 minutes.
An unmarked Martinez spurned a glorious chance to add a second when he scuffed a shot wide and Lionel Scaloni's side were made to pay for their profligacy when Chile levelled 12 minutes into the second half.
Vidal was awarded a spot-kick for a rash challenge from Nicolas Tagliafico after referee Wilmar Roldan checked the pitchside monitor and although the midfielder's penalty was brilliantly tipped against the crossbar by Martinez, Vargas nodded in following up.
Bravo palmed away Messi's drive 20 minutes from time and Gonzalez headed over the bar when his skipper presented him with another golden opportunity in a frustrating start for Argentina.