Atletico, which has won all its Europa League games since dropping out of the Champions League earlier this season, then made it 2-0 when France striker Griezmann grabbed his 17th goal of 2018.
The second legs will be played on April 12.
The Spain worldwide has since sat out LaLiga fixtures against Villarreal and Deportivo La Coruna but returned to training with the first-team squad earlier this week.
Koke's goal after 23 seconds set Europa League favourites Atletico Madrid on their way to a quarter-final first-leg on over Sporting Lisbon.
Atletico, the Europa League victor in 2010 and 2012, showed Sporting no mercy.
Both scorers took advantage of sloppy defending, with Diego Costa setting up Koke for the opener while Griezmann capitalised on an error by fellow Frenchman Jeremy Mathieu to fire a crisp low shot into the bottom corner.
In the last round Atletico steamrolled Lokomotiv with an aggregate 8-1 win, from five goals away at the RZD Arena, and the expectation is there to produce a similar result against their Portuguese opponents.
"We are going to put our hearts into it and play to win, to distance ourselves even more from third place and to be expectant of what Barcelona are doing".
On the club's badge, Juanfran added: "I am convinced that the club does the best for its Atlético people and the fans will support us".
The hosts led after eight minutes through captain Senad Lulic, but Valon Berisha equalised with a penalty. Two goals in three second-half minutes, from Felipe Anderson and Ciro Immobile, eventually cost Salzburg their 35-game unbeaten run.
RB Leipzig in a match with "Marseilles" for the first time in its short history has kept his goal intact, and the only goal, Timo Werner brought the Germans to the semifinals.