Brazil needed just 40 seconds to take the lead in Recife through Douglas Costa after the Uruguayan defence failed to deal with Willian's dangerous cross from the right.
The home side doubled their advantage just after the midway point of the first half when Renato Augusto capitalised on Alvaro Pereira's failed clearance to smash the ball into the net.
But Uruguay did not take long to pull one back, with Edinson Cavani converting from close range to breathe some life back into the game for the visitors.
Uruguay coach Oscar Tabarez took off Cristian Rodriguez for Alvaro Gonzalez at the break and they found themselves level four minutes into the second half.
Suarez was played through by Pereira and the nation's all-time leading goalscorer extended his record with an equaliser to ensure the points were shared.
After just 10 passes from kick-off Brazil had the lead, with Willian's dangerous cross eluding the Uruguay defence, allowing Bayern Munich's Costa to guide the ball home from close range.
Neymar had the opportunity to double the lead in the eighth minute after a well-worked free-kick, but his strike blasted into the side-netting from just outside the six-yard box.
Uruguay were hardly threatening early on, with a Suarez header from a free-kick the closest they came to getting an effort on goal, while Neymar had a tame shot saved by Fernando Muslera.
Brazil did eventually make it 2-0 in the 26th minute when Neymar played a cutting through ball to Renato Augusto, with the Beijing Guoan midfielder fooling Muslera with a dummy before smashing the ball into an empty net.
However, Uruguay struck back five minutes later, with Pereira nutmegging Willian before playing a cross to Carlos Sanchez, and the Monterrey winger headed the ball back for an unmarked Cavani to make it 2-1.
Neymar had a strike from distance fizz past the post, while Costa hit the side-netting as the hosts took a slender advantage into the break.
But their lead did not last long into the second half after Pereira played through Suarez, with the Barcelona forward tucking a shot under Brazil goalkeeper Alisson to make it 2-2.
While the chances petered out, the game began to heat up, with Suarez and Neymar among those shown yellow cards by Argentine referee Nestor Fabian Pitana.
Suarez had a chance to seal the three points late in a one-on-one with Alisson but the Internacional shot-stopper denied the 28-year-old with his leg, with the game ending all square.
Uruguay remain second in the 10-man group with 10 points, while Brazil move up to third.