Eren Derdiyok's second-half winner accounted for competition debutants Vitória SC and sent FC Basel 1893 through to the UEFA Champions League group stage for the first time since 2003.
Following a goalless first leg in Portugal two weeks ago, the Swiss Super League champions went in front on the night and on aggregate after eleven minutes when Valentin Stocker struck. However, within four minutes the visitors had a potentially decisive away goal as Fajardo converted a penalty following François Marque's foul on João Alves. Needing to score again to avoid elimination, Christian Gross's side duly delivered as Basel-born Derdiyok drove in a powerful shot after 54 minutes.
Basel had seized the initiative when a fine passing move unlocked the Guimarães defence, culminating in Ivan Ergić finding Stocker to put the hosts ahead. The Portuguese outfit were soon back on level terms, though, as Fajardo dispatched his spot-kick. And if Basel were the more attacking team for the remainder of the half, creating opportunities through Marcos Gelabert and Derdiyok, their efforts were in vain as Guimarães held firm.
Indeed, Manuel Cajuda's men were the more enterprising side after the restart with Yves Hadley Desmarets sending a shot narrowly wide of goalkeeper Franco Costanzo's left-hand post when well-placed. The Basel defence had cause for further consternation moments later as João Alves directed a dangerous header on target.
Perhaps realising their predicament, Basel raised their game and began to exert pressure on Guimarães. The increased tempo paid dividends when Derdiyok received a pass from strike partner Stocker and fired low past Nilson. A goal down, it was soon Guimarães who were doing the pressing and they thought they had equalised for a second time through substitute Roberto Calmon Felix three minutes from time – only to see the effort ruled out for offside.