The Scottish full-back did not feature in the UEFA Europa League clashes with Trencin after arriving from Dundee United, but capped his debut with a 12th-minute strike in Germany.
Hull started their final match of pre-season in emphatic fashion as David Meyler gave the visitors the lead in the fifth minute, with Robertson adding the second after Nikica Jelavic's effort had been parried by Sven Ulreich.
Stuttgart could have halved the deficit a minute before the half-hour mark when they were gifted a penalty after a reckless challenge from Maynor Figueroa on Martin Harnik.
However, Steve Harper produced an impressive save to his left to deny Vedad Ibisevic and maintain Hull's two-goal advantage.
The Bosnia-Herzegovina international, though, did get his name on the scoresheet nine minutes later as he collected a back-heeled pass before firing into the bottom corner out of Harper's reach.
Poor finishing let the Premier League side down in the second half, something that will worry Bruce.
Three minutes after the break, Jelavic released Tom Ince, but, having escaped the attentions of his marker, the former Crystal Palace winger lifted his shot high over the crossbar without testing Ulreich.
It was Paul McShane's turn to spurn a decent opportunity after Harper had made a good save to deny Timo Werner.
Sone Aluko found space behind the Bundesliga side's defence, but McShane was unable to convert his low pull back as Antonio Rudiger and Ulreich kept his efforts out.
Robert Snodgrass then inexplicably rolled an effort wide of an open goal as Hull continued to be wasteful in attack.
Bruce will now prepare his side for the opening weekend of the Premier League season, when Hull face QPR at Loftus Road, while Stuttgart are in DFB-Pokal action at Bochum next Saturday.
