The Barcelona midfielder was shown a straight red card towards the end of the first half following a confrontation with former Cardiff midfielder Gary Medel, who was also dismissed.
Messi, though, appeared to have been hard done by, with Chile defender Medel appearing to push his head towards the Argentina playmaker as they squared up, chest to chest.
Argentina, beaten by hosts Brazil in the semi-final, had taken an early lead through Manchester City forward Sergio Aguero.
Chile – whose semi-final defeat by Peru ended hopes of a third straight Copa America title – saw Manchester United forward Alexis Sanchez forced off with a suspected hamstring problem before Paulo Dybala made it 2-0 after 22 minutes
After Messi and Medel were dismissed by Paraguay referee Mario De Vivar, who had consulted a video replay of the incident, Arturo Vidal’s penalty pulled a goal back from Chile on the hour, with the VAR again called into action.
Aguero squandered a couple of late chances to put the result beyond doubt, but Argentina saw out the closing stages to secure third place as their trophy drought goes on.
Messi had made an early impact when his quick ball forwards from a free-kick just inside his own half put Aguero racing clear and the Manchester City striker went around the goalkeeper and opened the scoring in the 13th minute.
Chile were then forced into a change when Sanchez limped off and Junior Fernandes came on.
Argentina doubled their lead in the 22nd minute. Juventus forward Dybala was played into the left side of the Chile penalty area and clipped an angled shot into the far corner.
Dybala sent another volley just wide before tempers boiled over as half-time approached when Medel and Messi squared up after chasing a ball going out on the touchline.
Argentina remained a threat on the counter at the start of the second half, but Chile were given a way back into the match.
Play was called back as the referee looked at a potential foul by Argentina’s Giovani Lo Celso on Charles Aranguiz on the edge of the penalty area and eventually pointed to the spot.
Despite the delay, Barcelona midfielder Vidal remained calm, drilling the penalty straight down the middle as Argentina goalkeeper Franco Armani dived away.
Fernandes and fellow substitute Diego Vargas both tested Armani before Argentina slowly regained control, with Aguero having a couple of late chances to score again.
In stoppage time, the Chile bench were all up claiming for handball, but this time there was no extensive VAR review as the appeals were swiftly waved away.
Provided by Press Association Sport
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