Both runners appeared to get good jumps but Giants catcher Buster Posey picked the slower runner and threw a perfect laser to Joe Panik to get the quick tag of Morneau at second.
Weiss' challenge was for naught, as replay officials ruled the play would stand, and Brandon Barnes then grounded out to short to end the threat.
Coming in to protect a two-run lead, right-hander LaTroy Hawkins blew his second save of the season by serving up three consecutive hits to start the ninth.
Yet just as the caught-stealing attempt was looking to be extra costly, the Rockies rallied back for three singles in the ninth, including one of the walk-off variety from Charlie Blackmon that gave the Rockies a 10-9 victory.
Cody Ulm is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.