Davidson was pleased to progress 3-2 on penalties against the Championship side - with the scores 2-2 after extra-time - but admitted that Thursday evening's Europa League match against Galatasaray took a lot more out of his players than he thought.
Ricky Little missed the first penalty for Arbroath and their chances appeared to be over when Dale Hilson missed the target, but Jason Kerr and Callum Booth both failed to convert for Saints.
However, Bobby Linn hit the post for the hosts and Ali McCann wrapped up the Perth side's win.
Davidson said: "It was really difficult for us after playing at such a high intensity.
"It was a difficult place to play but full credit to my players as they never gave up despite Arbroath being in front twice during the game.
"I told the players at half-time to find a way to win the game and they did that so I am pleased with that. Cup competitions are important to us as a club and we are delighted to get through to the next round.
"We battled and could have done things a bit better but full credit to Arbroath as they made us work hard for that."
Arbroath took the lead on the half-hour mark with a dangerous corner from the right by Nicky Low giving St Johnstone trouble and Joel Nouble tapped the ball home from a few yards out.
St Johnstone levelled just before the hour mark after being awarded a penalty when Reece Devine was brought down inside the box. Kerr stepped up but was denied, although Glenn Middleton fired home the rebound from close range.
Arbroath went ahead for the second time just a couple of minutes into extra-time through Thomas O'Brien, who sweetly struck a volley into the roof of the net.
But Saints bounced back again after 105 minutes as the unmarked Jamie McCart guided home a fine header before the match went to penalties.
Arbroath manager Dick Campbell was happy with the effort put in by his players but was disappointed with their composure in front of goal during penalties.
Campbell said: "I am happy with the level of performance but raging with how the spot-kicks panned out - but it wasn't to be.
"We made St Johnstone not look at their best today and that's a pleasing factor for me. I enjoyed the experience but am just angry with how the game unfolded.
"We had chances to win the game both in normal time and in extra-time which we didn't take.
"Michael McKenna missed from six yards out and it was a golden opportunity. I wish St Johnstone all the best in the next round."