The 25-year-old's first-half try at the Galway Sportsground helped Sean Lineen's troops into a comfortable 10-3 lead before a second-half fightback, culminating in Niall O'Connor's late kick, saw the teams share the spoils.
Connacht, who had lost ten straight RaboDirect PRO12 games prior to the Glasgow clash, were benefitted from a number of errors and Welsh insists it will be back to the drawing board before a potentially decisive game against the European champions on Saturday.
"We were gutted not to get the win," said Welsh, whose try was his second in successive games.
"I felt as if it was there for us and we were let down by our own mistakes. We'll go back to the drawing board and we know what needs to be done and what needs to be fixed, so we'll prepare accordingly for Leinster next week and hopefully we can get back on the right track.
"Connacht got their kick in the last play of the game to get the draw and it was two points lost, that's definitely the feeling throughout the whole changing room and I've not felt like that throughout all of this season.
"We knew it was always going to be a tough game, every game in the RaboDirect PRO12 is, it's just extremely disappointing.
"Leinster is a big opportunity for us to get four points but I just hope that once it comes to the business end the Connacht result doesn't come back to haunt us. That's up to us, we know what needs to be done."
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