The former Millwall and MK Dons keeper sank to his knees before celebrating in the stands with his father, who played in the West Ham defence for 21 years.
"It still doesn't feel real," said Martin, who is a boyhood fan. "I'm just pleased I got through it, I didn't let anyone down and actually I enjoyed it in the end.
"My dad didn't say much - I think we were both in tears. For my dad to play for the club for 21 years and see it, and for me to make my debut, keep a clean sheet and get win for the boys, it is so pleasing.
I was struggling to eat for two days; Cressa [Cresswell] was laughing at me because my plate was taken away because I couldn't eat my food.
"I hope I've given myself a good chance [to keep my place] but I have to work hard and keep grounded.
West Ham boss Manuel Pellegrini: "It was very difficult [to drop Roberto]. It's a position you must be very careful of when you change it. I gave him the most amount of matches you can as a manager, but at the moment we needed a reaction.
"It was a risk with Martin because he had never played before in Premier League but he did very well. They are a complete team and the defenders helped him a lot to keep a clean sheet.
"It was a very important win, because we have had results that are not expected or deserved. You must be solid and consistent. That was the major change that we did compared with other games and it allowed us to keep a clean sheet and beat a team like Chelsea who are in a very good moment."
Chelsea head coach Frank Lampard: "We were not at our best, that was very clear. We came up below par and that's something I've not had to say much this season. If anything it's a test for us here and a reality check for us to react very quickly.
"We can't use playing in Valencia as an excuse, that's why we have a squad. We didn't create as much as we normally do, and we gave away too many opportunities. For a long time people have given us a lot of credit but football doesn't work like that, it's a test of character for us now.
"Tammy has been playing well and scoring goals, and when we lose him you're looking for goals not just in his area but different areas of pitch. We didn't have a enough punch in the final third in our game. We missed Tammy but that is not an excuse."