Martinez was making his home bow after replacing Kenny Jackett as manager last week, but even with Orient down to ten men after Adam Chambers' late sending off, good goalkeeping from Glyn Garner and missed chances stopped the Swans picking up a second win under the Spaniard's charge.
Martinez took the applause on his return but made no surprises in his line-up with the Swansea squad still decimated by injury.
Defender Alan Tate took over the armband duties with Darren Pratley ruled out for the season this week and stand-in Dennis Lawrence suspended.
Meanwhile, Orient kept the same side that saw off Tranmere a week earlier including hat-trick hero Ryan Jarvis.
But despite the expectation, there was little to really cheer in an even opening period until Lee Trundle's pass put Ian Craney clear on 13 minutes. But the midfielder, despite having two team-mates outside him, failed to take advantage of the chance and shot well wide to spare Orient.
Five minutes later and the hosts blew another good opportunity to break the deadlock. Trundle was again the provider with a cross but Izzy Iriekpen headed wide despite being unmarked inside the area.
Orient showed their teeth when a cutting run from Shane Tudor carved open the Swansea defence only to shoot tamely straight at Willy Gueret on 30 minutes.
Garner had to be on his toes two minutes later to keep out Andy Robinson's free-kick by tipping the ball around the post, before Chambers stung Gueret's palms with a sharp effort four minutes from time.
Trundle could have got the hosts off to the perfect second half start but he couldn't bury Leon Britton's through ball 12 minutes after the restart.
And try as they might, Swansea couldn't find a way to break the visitors down despite their dominance.
They were given a helping hand when Chambers deservedly saw red for a second booking 13 minutes from time, but Garner was equal to efforts from Tom Butler and Trundle before Gary Alexander cleared a Pawel Abbott effort off the line to earn a share of the spoils for the O's.