Both sides piled on the runs in the first innings but a West Indian second-innings collapse increased the prospects of an outright result.
Tino Best's first Test five-wicket haul ensured the home side never threatened to reach their 245-run target, victory being wrapped up with an hour to spare.
Sammy told reporters after the match: "We have said from the beginning, we had a lot of respect for Bangladesh and we were not going to take them lightly - and we didn't."
"Had we taken them for granted, the result of the first innings total would have caused panic in the dressing room."
He added: "But to see that we battled our way throughout the Test and came out victorious will definitely be a boost to the confidence that we already had coming out of the ICC World Twenty20 tournament.
"It will also help improve the attitudes of players whenever we take the field and face a difficult task."
Sammy believes the hosts will go into the second and final Test, starting on Wednesday in Khulna, with renewed confidence but his own team will be up for the challenge once more.
He said: "I know they will take a lot of positives from this game, but so will we. It leaves the series wide open and we have seen they are not a side that will roll over, so we expect a bigger fight and hopefully we can again be victorious."
