"We could have won, but now we have to correct our mistakes and think about Real Sociedad next weekend," said Barcelona defender Eric Garcia.
"We wanted to start the season on the right foot and with a victory, but we knew that Rayo gave us a lot of problems last year. They were good defensively tonight."
The hosts dominated the opening half an hour as Lewandowski was denied a goal on his competitive debut.
The Poland forward was judged to be offside from an Ousmane Dembele pass before his delicate chipped finish was cancelled out.
As rain fell at the Camp Nou in temperatures of 26 degrees Celsius (79 degrees Fahrenheit) Pedri came close to breaking the deadlock with 10 minutes of the half to play but his curling effort flew past the post.
Despite their first half control Marc ter Stegen ensured the sides stayed level at the break with a fine save to stop Rayo's Alvaro Garcia during injury time.
Barca continued their onslaught on the Rayo defence but were unable to beat goalkeeper Stole Dimitrievski, who was involved in two heavy collisions on his line within the first 10 minutes of the second half.
With half an hour to play Xavi brought on Frenkie de Jong for Gavi despite the Netherlands midfielder being linked with a move away from Catalunya to lighten club debt, which forced Lionel Messi to leave 12 months ago.
Falcao goes close
Xavi's Rayo counterpart Andoni Iraola also made changes in the final 30 minutes including giving experienced Colombia striker Radamel Falcao his competitive debut.
With nine minutes left De Jong combined with Lewandowski but the 33-year-old's shot skimmed the upright.
Seven minutes later another Barca summer signing was denied a goal as midfielder Franck Kessie, who arrived from AC Milan, was offside before beating Dimitrievski.
Xavi's side ended the game with 10 men as club skipper Busquets was red-carded for an elbow on Falcao in the fourth of eight injury time minutes.
Less than 60 seconds later, Falcao thought he had scored the winner, and claimed Rayo's third straight win over the Catalans, but was found to be offside and the visitors held on for the point.
Earlier, 21-year-old winger Alex Baena scored twice on just his eighth La Liga appearance as Villarreal beat Real Valladolid 3-0.
Forward Joselu claimed a point for Espanyol at Celta Viga with a 97th minute winner with a 2-2 draw after the Galicians' veteran striker Iago Aspas scored his side's opening goal of a season for a third straight campaign.
On Sunday, champions Real Madrid head to promoted Almeria while, on Monday, Atletico Madrid head south to face Getafe.
On Friday, Osasuna stunned Champions League qualifiers Sevilla 2-1 in the opening fixture of the campaign.