A counter-attack produced the opener for Uruguay, as Edinson Cavani met Suarez's cross to score from close range after 34 minutes.
Sergio Aguero gained the equaliser with a 63rd-minute header after connecting with Messi's free-kick.
However, the Uruguayans were back in the lead just five minutes later as Suarez curled a direct free-kick home.
Messi saved Argentina in stoppage time after Uruguay's Martin Caceres was penalised for handball in the box, and the Barcelona icon slotted away a penalty to tie the game.
Both teams started with strong line-ups for the friendly on foreign shores. Manchester City duo Aguero and Nicolas Otamendi started alongside Messi and Paulo Dybala, while Uruguay featured Diego Godin in defence.
Godin denied Dybala an early strike when the defender blocked the Juventus man's goal-bound effort.
Messi and Aguero appeared fluid and mobile in the Argentina attack, but Uruguay dealt with the best they could offer in the first half.
The deadlock was broken after Argentina were caught on their heels, allowing Cavani and Suarez to combine after a swift counter.
Lucas Torreira's delightful ball was passed across the box by Suarez, and Cavani made no mistake from six yards.
Argentina appeared frustrated going into half-time, but they were quickly into a positive rhythm after the interval.
An action-packed second half saw La Albiceleste claw their way back into the game with Messi as the instigator.
Aguero made no mistake from close range with a header from Messi's free-kick, with the Argentinians in the ascendancy.
However, Uruguay halted their opponents' rally cry in quick time when Suarez curled home a free-kick from outside the box.
The goal forced Messi and Co. into another gear, and Aguero should have grabbed his brace after combining with his captain.
Dybala was next to spurn a chance as Argentina chased a second equaliser, but the forward fluffed his lines and missed the target.
Argentina went for broke in the closing seconds and were rewarded when the referee judged Caceres had handled the ball in the penalty area.
The moment gave Messi the chance to score from the spot, and the attacker fired down the centre of the goal to level the scores in the 92nd-minute.