Mamardashvili suffered a wound as a result of a collision in a futile attempt to prevent Beto from equalising for Everton.
The hosts had drawn themselves level following an opener by Mohamed Salah during the first half.
“I’d be lying if I said I wasn’t nervous,” admitted Woodman. “Being a part of this massive club, you have moments where you are a little bit nervous.
“But, to be honest, I thought to myself as I ran on, I just saw our back four and thought, ‘I should be alright, you know, as they are proper players.’ I thought I wouldn’t be too busy and, to be fair, they looked after me.
“It was just unbelievable to make my debut in the Premier League and to win like we did was special.”