Carlo Ancelotti gave a number of players from Madrid’s B team a chance to shine in Saturday's match in California but Real took the lead through their world-record signing after just 10 minutes when Bale unleashed a ferocious, swerving shot from 25 yards which flew into the top corner.
Inter equalised midway through the second half when former Manchester United defender Nemanja Vidic was brought down in the box and Mauro Icardi stepped up to chip a cool penalty straight down the middle.
Neither side could force a winner and the tie was settled by spot-kicks, with Inter taking an early advantage after Isco’s effort was pushed on to the post by Juan Pablo Carrizo.
Asier Illarramendi and Omar Mascarell were also unsuccessful for Real and it was left to Icardi to convert the winning penalty to give Inter a winning start in Group A.
They now have two points in Group A and Real one after taking the match to penalties, while Manchester United top the table with three points after beating Roma 3-2.
Despite the spot-kick defeat, Ancelotti was content to see Bale continue to impress during his first pre-season proper with the club, having last year been hampered by a thigh injury.
"He had a really good holiday - 40 days - and he is fresh," Ancelotti said.
"He is working really well and doing what he was unable to do last season. Next season everyone is sure that we will be better for him (having had a full pre-season)."
Watch the International Champions Cup live on Sky Sports HD, with Manchester City taking on AC Milan and Liverpool facing Olympiakos on Sky Sports 1HD from 9pm on Sunday.