Norwich's goalless draw against Chelsea on Sunday did little to bolster their hopes of survival, leaving them two points adrift of safety and requiring a victory over Arsenal in their final game to stay up.
Even with a win over Arsene Wenger's men, Norwich will still likely require Sunderland to lose twice to maintain their top-flight status.
However, Snodgrass was positive after Norwich gave themselves a fighting chance and insists they will not give up.
"We wouldn't have been able to look at ourselves in the mirror if we hadn't given it our all [against Chelsea] and given ourselves the chance of taking it down to the very last day," he told the club's official website.
"We showed the character of men, we were leaders and we put our bodies on the line and tried to give everything we could.
"It's one of those games where you look back, and obviously you’ve got the point, but you're sitting there thinking 'Why are we in this position?' It's very frustrating.'"
The Scotland international, who joined from Leeds United in 2012, acknowledged they need all the help they can get from West Brom, who face Sunderland on Wednesday.
"We'll keep going, we'll keep believing that we can get something," he added. "It's out of our hands, so hopefully West Brom can do us a favour."