A stunning Craig Gilroy try and eight points from the boot of Paddy Jackson were enough for Ulster, but the visitors stayed in touch through Aly Muldowney's try and the boot of Jack Carty to set up a tense finish at the Kingspan Stadium.
But despite maintaining their excellent record against Connacht - Ulster having lost just once in 17 games in this fixture - and moving into the top four of the GUINNESS PRO12, Doak admitted he was just relieved to get over the line.
"We got the job done in the end," he said. "But obviously we're a little bit frustrated with the dominance we had up front and the fact we didn't get our just rewards.
"They're a dogged side and it was difficult playing conditions and it's always tight when you're up 10-3 and you've got dominance.
"We had a scrum and we were penalised 10 metres from their line and missed a couple of opportunities with kicks at goal and maybe if the chances had been taken it might have made it a little bit easier with 15 minutes to go on the old ticker.
"They did well to get into our 22 and get a good score in the second half, but to be fair to our guys coming in off the back of the two losses they showed great character."
Meanwhile Doak also added that Ruan Pienaar will have to wait a little longer to return for Ulster, as the fly-half's knee ligaments are still not fully recovered after four months on the treatment table already this season.
He added: "He tried running on it this week, we thought he might have made it, but the fluid in the knee is too much.
"We'll have to see how he goes in the next few days. I don't think it's as serious as the initial ligament damage, but when you have had an extended period of time out and get another bang to the same knee it can be a little bit daunting.
"It just wasn't right for him."
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