Keane opened the scoring in the third minute, beating Sturgis with a cut near the top of the box and finishing despite goalkeeper Tyler Deric's deflection.
Sturgis answered in the 11th minute for the Dynamo (1-1-1). David Horst headed Oscar Garcia's corner kick back into the middle of the 6-yard box and Sturgis ran underneath it for an easy finish.
The Galaxy (1-0-2) snapped a four-game regular-season losing streak to the Dynamo. Houston returned to the Western Conference this season after a four-year absence.
Deric preserved the draw with all three of his saves in the final 25 minutes, highlighted by a diving block in the 71st.
Dynamo midfielder Ricardo Clark left the game late in the first half with a chest injury.