Corey Baird had put the Dynamo, back in the playoffs for the first time since 2017, ahead in the 28th minute but Diego Luna levelled for Salt Lake in the 65th.
But Houston triumphed in the shoot-out with Griffin Dorsey sealing the victory with the final spot-kick to set up a clash with Sporting Kansas City in the next round.
MLS introduced 'best of three' series for the first round of the playoffs, a minimal version of the format used by the NBA and Major League Baseball.
But the remainder of the tournament is now single-elimination games and promises the kind of do-or-die drama that the crowd at Shell Energy Stadium enjoyed.
Dorsey had a great chance to open the scoring in the 27th minute, picking the ball up on the right and surging into space with the RSL defence backing off but he put his tame shot wide.
The breakthrough came a minute later though when a Nelson Quinones cross from the left was headed back across the box by Hector Herrera and Baird poked home.
The Dynamo were unable to build on that advantage though and Salt Lake fought back after the break when Bryan Oviedo whipped in a cross from the left which was met with a firm header from the unattended Luna.
Houston came closest to securing a win in regulation - Quinones worked himself an opening down the left but then shot meekly straight at RSL keeper Zac MacMath.
With two minutes left on the clock, French midfielder Amine Bassi's fierce shot was parried out by MacMath who then recovered to deal with Thorleifur Ulfarsson's follow-up header.
Game Two had been won by RSL on penalties but this time they trailed after Luna's shot was saved by Houston keeper Steve Clark and Bassi converted to make it 3-2 for the Dynamo.
A soft penalty, easily saved by McMath, levelled up the shoot-out however but Salt Lake sub Daniel Musovski then hit the post, leaving Dorsey to tuck his shot into the corner and set off the celebrations.
On Friday, the Seattle Sounders won their Game Three tie with FC Dallas and they will face defending champions Los Angeles FC in the next round.
The first round closes on Sunday with Columbus Crew hosting Atlanta United in the East with Orlando City awaiting in the conference semi.
The other Eastern Conference semi-final will see top-seed FC Cincinnati take on last season's runners-up Philadelphia Union.
