The build-up to Monday's Chile-Uruguay game had been overshadowed by controversy in the Chile camp after revelations the team had breached Covid-19 protocols by bringing a local hairdresser into the team hotel.
CONMEBOL said it would fine those involved $30,000.
However there was no sign of the controversy affecting their play, with Vargas firing Chile ahead on 26 minutes from a tight angle after latching onto a clever pass from English-born Ben Brereton.
Uruguay grabbed a share of the points in the second-half when Suarez bundled in the ball from close range during a tangle with former Barcelona team-mate Arturo Vidal at the far post following a corner.
Manchester United striker Edinson Cavani almost snatched victory for Uruguay in the 85th minute, but his glancing header flashed just wide of the post.
Vargas's first-half strike saw him draw level with Peru's Paolo Guerrero on 14 goals in the Copa America's all-time goalscoring rankings, three behind the record of 17 held by Brazil's Zizinho and Argentina's Norberto Mendez.