Lionel Messi came on as a substitute shortly before Alcacer's scrambled a 64th-minute goal and at the same time as Sergi Roberto was introduced, but the latter was sent off for hauling back Iago Aspas.
Spain striker Aspas smuggled home Celta's second equaliser eight minutes from time and the hosts pressed hard for a third win over Barca at Balaidos in as many seasons.
"Celta always generate many opportunities and play with many people up [in attack]," Suarez told reporters.
"I think in the second half we dominated until the red card. And it is true that they had several [chances] to score the third.
"Luckily it was not like that and we are still unbeaten and a little closer to [the] La Liga [title]. We have to give ourselves credit because to continue unbeaten at this point is an incredible achievement."
Suarez was making his first top-flight start since the turn of the year and added: "When so much time goes by is always difficult. My goal is to succeed here and keep trying."