Gary Fraser's superb early strike had Thistle on course for a second Premiership win in four days before Craig Wighton pulled the home side level and Jags keeper Scott Fox was sent off.
In the end Firhill gaffer Archibald was pleased to see his side dig in to maintain their unbeaten start to the campaign.
He said: "I have to give the lads a lot of credit. Under the circumstances, it's a good point. You never know what would have happened if Scott hadn't been sent off, but that's football.
"I was really pleased with our performance in the first half. We were fantastic. That's how we want to play every week.
"We pressed the ball, moved it really well and created a number of chances but just didn't take them - apart from Gary's wonder goal.
"We can take heart and go into the next game hopeful of getting another positive result."
Thistle have yet to decide whether they will appeal the decision by ref Craig Charleston that a clash between Fox and Dundee forward Paul McGowan was violent conduct meriting a red card and a penalty.
Former Scotland goalie Paul Gallacher came on as substitute, with his first act being to save Gary Harkins' spot-kick to keep Thistle on level terms.
Afterwards the Partick boss hailed his back-up keeper and coach, who also celebrated his 35th birthday on Saturday.
Archibald said: "Paul did fantastically well. He had two great saves. First of all the penalty, then he got down well for a header.
"We played with 10 men for half an hour, so we showed a bit of grit and determination.
"It's been a good start for us but we know it could have been better if we'd taken our chances."