Anichebe had a golden chance to win it for the Blues in the dying minutes, only to be denied by a fine smothering save from goalkeeper Hugo Lloris.
"I didn't realise he was that quick," he told evertontv. "I was thinking, 'should I have taken it sooner'. I tried to chip it in the end and it wasn't meant to be.
"I'm happy though. I thought I had a good game. It would have been nice to round it off with a goal at the end but at least I was there to try to score and hopefully from now until the end of the season I can keep getting more chances, keep playing well - that's the most important thing - and hopefully get on the scoresheet too."