Suarez struck with virtually the last touch of the first half to bring his side level after Paolo Guerrero had put Peru ahead.
Diego Forlan then had the chance to win it for Oscar Tabarez's men, but the Atletico striker was off target when it would have been easier to score.
After a slow start, Peru brought the game to life in the 23rd minute when Guerrero sprung the offside trap and raced half the length of the field to round Fernando Muslera and open the scoring.
Maxi Pereira had a shot deflected wide as Uruguay sought a way back into the match, and Forlan thought he had it when he curled a free-kick towards the top corner, only to see Raul Fernandez claw it away.
But Suarez did the business when he found himself one-on-one with the stopper just before the half-time whistle, slotting coolly under his body for an 18th goal for his country.
Juan Manuel Vargas dragged a shot just wide for Peru after the break, before Suarez played Forlan through on goal, only for the normally accurate striker to sidefoot high and wide.
Vargas then saw another effort drift wide and Guerrero had more time than he realised with a free, last-ditch header as the match ended in a draw, the fourth in five games so far in the tournament.