Gardner, 22, graduated from Villa's youth academy in 2011 before making his first-team debut later that year.
Meanwhile, 19-year-old Robinson made his breakthrough for the Villans last season.
"I've been here from a young age and I've been to Villa games all my life as a fan so it is a dream come true to play for the first team like I have done," Gardner told the club's official website.
Robinson added: "I am absolutely delighted to sign a new deal here. Now it's about pushing on and progressing.
"Last season was fantastic. I didn't expect so many positives to come out of it. To work as hard as I have and then have the gaffer give me a chance with the first team, it was amazing. I have loved every minute of it."
Villa start the new Premier League season at Stoke City on August 16.