The Welshman had put Stuart Pearce's side ahead in the 20th minute but the hosts tired as the game went on and Moussa Konate broke through to equalise late in the game.
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"I just got caught and tweaked my medial ligament a little bit, but I should be alright," he told reporters after the game. “But we drew due to Senegal equalising, not due to tackles.
"I felt physically quite good, but I have to realise that I am only two weeks into pre-season - I think you could see in the last 20 minutes that it caught up with us a little bit.
"But the great thing about football is that we have another game coming up soon and we will get fitter as this tournament goes on. We have to stay in this tournament."
Particularly before his goal, the Liverpool forward was frequently booed by Manchester United-supporting local fans, but the 33-year-old was ready, after seeing Reds colleague Luis Suarez receive similar treatment while playing for Uruguay earlier that day.
"It's very rare [to get a standing ovation]," Bellamy noted. "I was expecting what Luis got earlier so I was sort of prepared. I was grateful that Luis was playing first.
"But this is the Theatre of Dreams. I'm a Liverpool fan and Liverpool is my club, but it's always an honour to play at Old Trafford. It's a truly remarkable place to play football and to come here and represent Great Britain and my own country Wales was very special.
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"You know I can make mistakes but when I play for a team I work as hard as I can for a team - hopefully I got recognition for that today.
"I just tried to keep it down," he added of his first-half strike. "I didn't just want to sky it, I just wanted it down. It ended up a bit too far down for my liking but a goal is a goal and to get the first goal in this tournament for Team GB is definitely special.
"It is important to try and win the first group game; we've seen Uruguay do that [2-1 against UAE].
"We wanted to start well and unfortunately we didn't, but I'm getting more confident as we go along.
"We have to win against UAE and we're going to give it a really good go at another remarkable stadium. I should be fine [for Sunday]."