The result, just the fourth time an MLS team have managed a draw on Mexican soil in an official game, gives Salt Lake a good chance of becoming the first team from the league to gain a place in FIFA's World Club Cup - the prize for winning the club tournament for North and Central America and the Caribbean.
However, Salt Lake will have to take on the Mexican champions in next Wednesday's game in Utah without their influential skipper Kyle Beckerman, who will be suspended after picking up a second yellow card.
Monterrey grabbed the lead in a pulsating encounter with an 18th minute strike from close-range by forward Aldo de Nigris, after Real failed to clear a goalmouth scramble.
Surprisingly, Monterrey coach Victor Manuel Vucetich subbed De Nigris and captain Luis Ernesto Perez almost immediately. While both players used ice packs on their legs sitting on the bench neither looked seriously injured.
Although Real played their usual short-passing game, it was a more direct method that brought them back on level terms in the 35th minute when Will Johnson curled in a cross from deep on the right and Nat Borchers rose well to head home.
The Mexicans came out strongly after the break, gaining the upper hand in midfield and they restored their advantage with a 63rd minute penalty from Chilean forward Humberto Suazo after defender Jamison Olave was ruled to have handled in the area.
With Beckerman's booking delivering another blow to the visitors, the tie looked to be in Monterrey's control but they may live to regret a missed chance by Suazo, who raced into the area but lobbed his shot wide over the advancing keeper Nick Rimando.
With just a minute of normal time remaining, substitute Arturo Alvarez won possession for Salt Lake and jinked into space before delivering a superb pass to Argentine Morales, who cut into the area and drove a fierce shot low into the far corner for a dramatic equaliser and a valuable second away goal.
Real Salt Lake have an excellent record at their Rio Tinto stadium and are unbeaten in their last 34 regular season and Cup games.
