Scotland created the better chances against the second seeds in Group A but a section of the Hampden Park crowd booed their team at the final whistle.
Steven Naismith had the best opportunity when he was played through by Gary Caldwell, but missed the target with only the goalkeeper to beat. Kenny Miller failed to make the most of several half-chances and Robert Snodgrass also had a one-on-one, saved well by Vladimir Stojkovic from a tight angle.
Scotland goalkeeper Allan McGregor made a vital last-ditch save from Dusan Tadic in the final minute, but even then Scotland had the last chance, with Stojkovic saving well from substitute James Forrest.
The Wolverhampton Wanderers defender said: "I thought we had the better chances in the game, Greegs only had one save to make towards the end. We're disappointed not to get the three points but we'll take a point and take a lot of positives from the game and we're looking forward to Macedonia on Tuesday.
"I thought from back to front we defended well and were solid. They had a lot of the ball at some points during the game but we frustrated them and had chances on the counter-attack."
Berra accepted that Scotland fans will expect to win home games, which continue with the visit of FYR Macedonia. But he added: "I think the way we performed against Serbia, who are a top team, when we go over there to play them we won't have any fears.
"We deserved to win the game, we had the better chances and we were just unlucky the ball didn't fall for us in the box a couple of times. If we win on Tuesday we walk away with four points and I think that would be pleasing all round."
'Great' debut by Dixon
Manager Craig Levein had several worries over his defence earlier in the week, with a total of five full-backs missing. But Huddersfield Town left-back Paul Dixon stepped in for an impressive debut.
Berra said: "A lot of people might have had doubts but he was top man, a well-deserved man of the match and put in a great performance. He defended one-on-ones very well against [Branislav] Ivanovic and going forward he held his own and put some good crosses in. I don't think you could ask for a better debut."